◀ ▲ ▶Index / Index: By Contributors. Thank you!
Index: By Contributors. Thank you!
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- bookofproofs (8396)
- Branches
- History
- Algorithm: Addition (or Subtraction) of Two Registers
- Algorithm: Assignment of a Constant `\(c\)` to the Register `\(r_i\)` with a `\(L O O P\)`-Program
- Algorithm: Calculation of Inverses Modulo a Number (Python)
- Algorithm: Complex Numbers
- Algorithm: Conditional execution of `\(L O O P\)` programs - IF-THEN construct
- Algorithm: Conditional execution of `\(L O O P\)` programs - IF-THEN-ELSE construct
- Algorithm: Continued Fraction (Python)
- Algorithm: Converting Decimal Numbers to Roman Numbers
- Algorithm: Converting Roman Numbers to Decimal Numbers
- Algorithm: Division with Quotient and Remainder (Unit-cost Random Access Machine)
- Algorithm: Extended Greatest Common Divisor (Python)
- Algorithm: Get All Components of a Graph
- Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex
- Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph
- Algorithm: Graph (Python)
- Algorithm: Greatest Common Divisor (Python)
- Algorithm: Horner Scheme
- Algorithm: Jacobi Symbol (Python)
- Algorithm: Multiplication of Two Registers
- Algorithm: No-Operation Command (NOP)
- Algorithm: Sequential Search
- Algorithm: Setting the value of a register to the value of another register plus or minus a constant
- Algorithm: getSubgraph
- Application: Application of Monotonic Convergence (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
- Application: Application of the Rabin-Scott Theorem - the Rabin-Scott Algorithm (related to Chapter: Finite Automata (Finite Sequential Machines))
- Application: Application of the Theorem to Reduce `$\epsilon$`-NFA to NFA (related to Chapter: Finite Automata (Finite Sequential Machines))
- Application: Lightning and Thunder or the Relativity of Acoustic Signals (related to Part: Oddities and Curiosities)
- Application: SLEs Revised in the Light of Vector Spaces (related to Chapter: Vectors Revised - Vector Spaces)
- Axiom: "Between" Relation, Axioms of Order
- Axiom: 1.1: Straight Line Determined by Two Distinct Points
- Axiom: 1.2: Segment Extension
- Axiom: 1.3: Circle Determined by its Center and its Radius
- Axiom: 1.4: Equality of all Right Angles
- Axiom: 1.5: Parallel Postulate
- Axiom: Addition of the Probability of Mutually Exclusive Events
- Axiom: Archimedean Axiom
- Axiom: Axiom of Choice
- Axiom: Axiom of Distributivity
- Axiom: Axiom of Empty Set
- Axiom: Axiom of Existence
- Axiom: Axiom of Extensionality
- Axiom: Axiom of Foundation
- Axiom: Axiom of Infinity
- Axiom: Axiom of Pairing
- Axiom: Axiom of Power Set
- Axiom: Axiom of Replacement (Schema)
- Axiom: Axiom of Union
- Axiom: Axioms of Connection
- Axiom: Axioms of Group
- Axiom: Axioms of Magma
- Axiom: Axioms of Monoid
- Axiom: Axioms of Semigroup
- Axiom: Bivalence of Truth
- Axiom: Filter
- Axiom: Peano Axioms
- Axiom: Principle of Relativity
- Axiom: Principle of the Constancy of Light Speed
- Axiom: Probability as a Non-Negative Number
- Axiom: Probability of the Certain Event
- Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension)
- Axiom: Separation Axioms
- Axiom: Zermelo-Fraenkel Axioms
- Branch: Algebra
- Branch: Analysis
- Branch: Combinatorics and Discrete Mathematics
- Branch: Game Theory
- Branch: Geometry
- Branch: Graph Theory
- Branch: Introduction to the Axiomatic Method
- Branch: Knot Theory
- Branch: Logic
- Branch: Number Systems and Arithmetics
- Branch: Number Theory
- Branch: Probability Theory and Statistics
- Branch: Riddles, Puzzles and Brain-Teasers
- Branch: Set Theory
- Branch: Theoretical Computer Science
- Branch: Theoretical Physics
- Branch: Topology
- Chapter: Abel Integral Equations
- Chapter: Affine Spaces
- Chapter: Algebraic Properties of Complex Numbers
- Chapter: Algebraic and Transcendent Numbers
- Chapter: An Introduction to the Principle of Duality in Projective Geometry
- Chapter: And / Or Trees
- Chapter: Applications of the Fundamental Theorem of the Difference Calculus
- Chapter: Arithmetic Functions
- Chapter: Arithmetical and Algebraic Problems
- Chapter: Around 1600 - The homogeneity principle and the birth of the mathematical formula
- Chapter: Avoidance of Negative Solutions, Descartes' Analysis and Synthesis
- Chapter: Banach Spaces and Banach Algebra
- Chapter: Basic Geometric Concepts in Projective Geometry
- Chapter: Basic Topological Concepts
- Chapter: Basics of Real Analysis of One Variable
- Chapter: Binary Relations and Their Properties
- Chapter: Bipartite Graphs
- Chapter: Branch and Bound
- Chapter: Calculation of Roots
- Chapter: Calendar
- Chapter: Can Cardinals be Ordered?
- Chapter: Chebyshev's Lemma
- Chapter: Chessboard Problems
- Chapter: Classification of Differential Equations
- Chapter: Classification of Formal Languages
- Chapter: Classifying the Logical Calculi - Variables, Formulas, Predicates and Signatures
- Chapter: Colorings of Planar Graphs
- Chapter: Combination and Group Problems
- Chapter: Combinatorial Algorithms
- Chapter: Compact Sets
- Chapter: Completeness of Real Numbers
- Chapter: Conditions for Planarity and Planarity Testing
- Chapter: Congruence Classes and Modular Arithmetic
- Chapter: Construction of Topologies
- Chapter: Continuity
- Chapter: Continuous Time Markov Chains
- Chapter: Continuum Hypothesis
- Chapter: Contraction and Minors
- Chapter: Contradictory Propositions in Propositional Logic
- Chapter: Correctness
- Chapter: Criteria for Convergence of Sequences
- Chapter: Crossing River Problems
- Chapter: Cryptography
- Chapter: Data Compression
- Chapter: Decidability
- Chapter: Density and Countability
- Chapter: Determinants
- Chapter: Differentiability
- Chapter: Differential Operations on Vector Fields
- Chapter: Differentiation
- Chapter: Digraphs
- Chapter: Discrete Fourier Transform (DFT)
- Chapter: Discrete Time Markov Chains
- Chapter: Divisibility
- Chapter: Divisibility and Modular Arithmetic
- Chapter: Divisibility in General Rings
- Chapter: Dynamic Programming
- Chapter: Eigenvalues and Eigenvectors
- Chapter: Eigenvalues and Eigenvectors
- Chapter: Elementary Results About Prime Numbers
- Chapter: Entropy
- Chapter: Equivalent Transformations in Propositional Logic
- Chapter: Error Analysis
- Chapter: Euclid's “Elements”
- Chapter: External Sorting
- Chapter: Extrapolation
- Chapter: Fields (Overview)
- Chapter: Files
- Chapter: Finite Automata (Finite Sequential Machines)
- Chapter: Fisher Distribution
- Chapter: Fixed Point Theory
- Chapter: Fourier Series
- Chapter: Fractional Arithmetic
- Chapter: Fredholm Integral Equations
- Chapter: From the Syntax to the Semantics of Formal Languages
- Chapter: Functions (Maps)
- Chapter: Fuzzy Logic
- Chapter: Genetic Programming
- Chapter: Geometric Aspects
- Chapter: Geometrical Problems
- Chapter: Goldbach Conjecture
- Chapter: Graph Representations
- Chapter: Graph-Theoretic Algorithms
- Chapter: Graphs
- Chapter: Graphs
- Chapter: Greedy Algorithms
- Chapter: Group-theoretic Problems
- Chapter: Groups (Overview)
- Chapter: Hashing
- Chapter: Hilbert Spaces
- Chapter: Hilbert's Axiomatic System
- Chapter: Hypergeometric Distribution
- Chapter: Important Properties of Binary Operations
- Chapter: Infinite Products - Overview
- Chapter: Infinite Series - Overview
- Chapter: Infinite Series Approximations
- Chapter: Integrability
- Chapter: Integral Operations on Vector Fields
- Chapter: Integration
- Chapter: Integration of Starting Value Problems (Ordinary DE)
- Chapter: Interpolation
- Chapter: Introduction to Matrices and Vectors
- Chapter: Invalid Logical Arguments
- Chapter: Isolated Singularities
- Chapter: Isomorphism (Graphs)
- Chapter: Jordan Normal Form
- Chapter: Laplace Probability
- Chapter: Laurant Series
- Chapter: Lebesgue Measure and Lebesgue Integral
- Chapter: Linear Equation Systems
- Chapter: Linear Equations and Systems of Linear Equations (SLEs)
- Chapter: Linear Operators and Linear Functionals
- Chapter: Linear Programming
- Chapter: Linked Lists
- Chapter: Liouville-Neumann Series
- Chapter: Logarithmic Normal Distribution
- Chapter: Logical Arguments Used in Mathematical Proofs
- Chapter: Magic Square Problems
- Chapter: Magmas, Semigroups, Monoids (Overview)
- Chapter: Mazes and How to Thread Them
- Chapter: Measuring, Weighing, and Packing Problems
- Chapter: Meromorphic Functions
- Chapter: Methods to Solve Ordinary DE
- Chapter: Methods to Solve Partial DE
- Chapter: Min-Max and Alpha-Beta
- Chapter: Mixing
- Chapter: Models of Computation
- Chapter: Modules (Overview)
- Chapter: Moving Counter Problems
- Chapter: Multiplication
- Chapter: Network Flows
- Chapter: Network Paths
- Chapter: Neural Networks
- Chapter: Non-Linear Equation Systems
- Chapter: Non-linear Operator
- Chapter: Normal Distribution
- Chapter: Normal Forms in `$PL0$`
- Chapter: Order
- Chapter: Order Relations
- Chapter: Orthogonalisation
- Chapter: Peano Arithmetic
- Chapter: Points and Lines Problems
- Chapter: Poisson Distribution
- Chapter: Polynomial Rings, Irreducibility, and Field Extensions
- Chapter: Positional Notation Systems
- Chapter: Power Series Introduction
- Chapter: Preface
- Chapter: Prime Number Theorem
- Chapter: Prime Number Theorem for Arithmetic Progressions
- Chapter: Principal Axis Transformation
- Chapter: Problems Concerning Games
- Chapter: Product Manipulation Methods
- Chapter: Properties of Complex Functions
- Chapter: Properties of Real Functions
- Chapter: Pseudo-Random Numbers
- Chapter: Putting it All Together - Syntax and Semantics of a Logical Calculus
- Chapter: Puzzle Games
- Chapter: Quadratic Residues
- Chapter: Quantum Computation
- Chapter: Queues
- Chapter: Random Sample Models
- Chapter: Randomized Rounding
- Chapter: Real Numbers As Limits Of Rational Numbers
- Chapter: Real-valued Sequences and Limits of Sequences and Functions
- Chapter: Reduction
- Chapter: Renaissance and Beginnings of the Infinitesimal Methods
- Chapter: Representation of Functions as Taylor Series
- Chapter: Residue Theorem
- Chapter: Rings (Overview)
- Chapter: Roman Numbers
- Chapter: Rotations and Basic Transformations
- Chapter: Satisfaction Problem Solving
- Chapter: Searching
- Chapter: Sequences and Limits
- Chapter: Sequences of Numbers
- Chapter: Set Operations
- Chapter: Sets
- Chapter: Sieve Methods
- Chapter: Similarity of Graphs
- Chapter: Simple Connectivity
- Chapter: Simple Facts Regarding Cardinals
- Chapter: Simple Formulas Involving the Complex Numbers
- Chapter: Simulated Annealing
- Chapter: Solving Diophantine Equations
- Chapter: Some Important Constants
- Chapter: Sorting
- Chapter: Stacks
- Chapter: String Matching
- Chapter: Strings
- Chapter: Strong Law of Large Numbers
- Chapter: Student Distribution
- Chapter: Sum Manipulation Methods
- Chapter: Symmetry Groups
- Chapter: The Paradox Party
- Chapter: The Proving Machine - an Automation of Logical Reasoning
- Chapter: The `$C(X)$` Algebra
- Chapter: The `$\chi^2$` Distribution
- Chapter: Topological Aspects
- Chapter: Trees
- Chapter: Triangle of the Stirling Numbers of the First Kind
- Chapter: Triangle of the Stirling Numbers of the Second Kind
- Chapter: Types of Complex Functions
- Chapter: Types of Real Functions
- Chapter: Unclassified Problems
- Chapter: Unicursal and Route Problems
- Chapter: Uniform Distribution
- Chapter: Useful Inequalities
- Chapter: Vector Properties of Complex Numbers
- Chapter: Vector Spaces (Overview)
- Chapter: Vectors
- Chapter: Vectors Revised - Vector Spaces
- Chapter: Volterra Integral Equation
- Chapter: Weak Law of Large Numbers
- Chapter: Zeta Functions
- Corollary: (Real) Exponential Function Is Always Positive (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
- Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
- Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle)
- Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Corollary: 5.07: Ratios of Equal Magnitudes (related to Proposition: 5.07: Ratios of Equal Magnitudes)
- Corollary: 5.19: Proportional Magnitudes have Proportional Remainders (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
- Corollary: 6.08: Geometric Mean Theorem (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
- Corollary: 6.19: Ratio of Areas of Similar Triangles (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
- Corollary: 6.20: Similar Polygons are Composed of Similar Triangles (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
- Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
- Corollary: 8.02: Construction of Geometric Progression in Lowest Terms (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
- Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II)
- Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero. (related to Corollary: `\(0x=0\)`)
- Corollary: Abelian Group of Vectors Under Addition (related to Proposition: Abelian Group of Matrices Under Addition)
- Corollary: Algebraic Structure of Strings over an Alphabet (related to Definition: Strings (words) over an Alphabet)
- Corollary: All Boolean Functions Can Be Built Using Conjunction, Disjunction, and Negation (related to Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms)
- Corollary: All Uniformly Continuous Functions are Continuous (related to Definition: Uniformly Continuous Functions (Real Case))
- Corollary: All Zeros of Cosine and Sine (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Arguments for which Cosine and Sine are Equal to Each Other (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Barycentric Coordinates, Barycenter (related to Definition: Affine Basis, Affine Coordinate System)
- Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: Bounds for the Minimal Tree Decomposability (related to Definition: Minimal Tree Decomposability)
- Corollary: Cartesian Products of Countable Sets Is Countable (related to Proposition: Union of Countably Many Countable Sets)
- Corollary: Circular References Of Self-Contained Sets Are Forbidden (related to Axiom: Axiom of Foundation)
- Corollary: Closed Real Intervals Are Compact (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
- Corollary: Commutativity of Conjunction (related to Definition: Conjunction)
- Corollary: Commutativity of Disjunction (related to Definition: Disjunction)
- Corollary: Commutativity of Equivalence (related to Definition: Equivalence)
- Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
- Corollary: Continuous Real Functions on Closed Intervals are Bounded (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
- Corollary: Contraposition of Cancellative Law for Adding Natural Numbers (related to Proposition: Addition of Natural Numbers Is Cancellative)
- Corollary: Convergence of Complex Conjugate Sequence (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
- Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Corollary: Cor. 10.009: Commensurability of Squares (related to Proposition: Prop. 10.009: Commensurability of Squares)
- Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
- Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
- Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)
- Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
- Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
- Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
- Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
- Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
- Corollary: Cosine and Sine are Periodic Functions (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Derivative of a Constant Function (related to Definition: Constant Function Real Case)
- Corollary: Derivative of a Linear Function `\(ax+b\)` (related to Definition: Linear Function)
- Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Differentiable Functions are Continuous (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
- Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions (related to Proposition: Diophantine Equations of Congruences)
- Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor)
- Corollary: Equality of Sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III)
- Corollary: Estimating the Growth of a Function with its Derivative (related to Theorem: Darboux's Theorem)
- Corollary: Even Number of Vertices with an Odd Degree in Finite Digraphs (related to Lemma: Handshaking Lemma for Finite Digraphs)
- Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs (related to Lemma: Handshaking Lemma for Finite Graphs)
- Corollary: Every Distance Is Positive Definite (related to Definition: Metric (Distance))
- Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I)
- Corollary: Every uniformly convergent sequence of functions is pointwise convergent. (related to Definition: Pointwise and Uniform Convergence)
- Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments (related to Definition: Continuous Functions at Single Real Numbers)
- Corollary: Existence of Arbitrarily Small Positive Rational Numbers (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Arbitrarily Small Powers (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle) (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers) (related to Definition: Multiplication of Natural Numbers)
- Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers) (related to Proposition: Addition Of Natural Numbers)
- Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Corollary: Existence of Powers Exceeding Any Positive Constant (related to Axiom: Archimedean Axiom)
- Corollary: Existence of Unique Integers Exceeding Real Numbers (related to Axiom: Archimedean Axiom)
- Corollary: Exponential Function Is Non-Negative (Real Case) (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Exponential Function Is Strictly Monotonically Increasing (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Exponential Function and the Euler Constant (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point (related to Definition: Continuous Functions at Single Real Numbers)
- Corollary: General Associative Law (related to Definition: Associativity)
- Corollary: General Associative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Associative)
- Corollary: General Commutative Law (related to Definition: Commutativity)
- Corollary: General Commutative Law of Multiplication (related to Proposition: Multiplication of Real Numbers Is Commutative)
- Corollary: Intersection of Convex Affine Sets (related to Definition: Convex Affine Set)
- Corollary: Irrational Numbers are Uncountable (related to Proposition: Real Numbers are Uncountable)
- Corollary: Justification of Power Set (related to Axiom: Axiom of Power Set)
- Corollary: Justification of Set Intersection (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Justification of Set Union (related to Axiom: Axiom of Union)
- Corollary: Justification of Subsets and Supersets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Justification of the Difference (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Justification of the Set-Builder Notation (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Limit of N-th Roots (related to Proposition: Rational Powers of Positive Numbers)
- Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets (related to Axiom: Axiom of Infinity)
- Corollary: Monotony Criterion for Absolute Series (related to Proposition: Monotony Criterion)
- Corollary: More Insight to Euler's Identity (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Negative Cosine and Sine vs Shifting the Argument (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Corollary: Non-Cauchy Sequences are Not Convergent (related to Theorem: Completeness Principle for Real Numbers)
- Corollary: Number of Vertices, Edges, and Faces of a Dual Graph (related to Definition: Dual Planar Graph)
- Corollary: Order Relation for Natural Numbers is Strict Total (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
- Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Planarity of Subdivisions (related to Definition: Subdivision of a Graph)
- Corollary: Primality of the Smallest Non-Trivial Divisor (related to Proposition: Existence of Prime Divisors)
- Corollary: Prime Dividing Product of Primes Implies Prime Divisor (related to Lemma: Generalized Euclidean Lemma)
- Corollary: Probability of Laplace Experiments (related to Definition: Laplace Experiments and Elementary Events)
- Corollary: Probability of the Impossible Event (related to Proposition: Probability of the Complement Event)
- Corollary: Properties of Transitive Sets (related to Definition: Transitive Set)
- Corollary: Properties of a Real Scalar Product (related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case))
- Corollary: Properties of the Absolute Value (related to Definition: Absolute Value of Real Numbers (Modulus))
- Corollary: Real Numbers Can Be Approximated by Rational Numbers (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
- Corollary: Real Polynomials of Odd Degree Have at Least One Real Root (related to Proposition: Limits of Polynomials at Infinity)
- Corollary: Reciprocity Law of Falling And Rising Factorial Powers (related to Definition: Falling And Rising Factorial Powers)
- Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Complex Exponential Function)
- Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function of General Base)
- Corollary: Reciprocity of Exponential Function, Non-Zero Property (related to Proposition: Functional Equation of the Exponential Function)
- Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Reduction of `$\epsilon$`-NFA to DFA (related to Theorem: Reduction of `$\epsilon$`-NFA to NFA)
- Corollary: Representing Real Cosine by Complex Exponential Function (related to Definition: Cosine of a Real Variable)
- Corollary: Representing Real Sine by Complex Exponential Function (related to Definition: Sine of a Real Variable)
- Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rules for Exponentiation in a Group (related to Definition: Exponentiation in a Group)
- Corollary: Rules of Calculations with Inequalities (related to Definition: Order Relation of Real Numbers)
- Corollary: Set Difference and Set Complement are the Same Concepts (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Simple Conclusions For Multiplicative Functions (related to Definition: Multiplicative Functions)
- Corollary: Solutions of a Linear Equation with many Unknowns (related to Definition: Linear Equations with many Unknowns)
- Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Corollary: Strictly, Well-ordered Sets and Transitive Sets (related to Theorem: Mostowski's Theorem)
- Corollary: Sufficient Condition for a Function to be Constant (related to Corollary: Estimating the Growth of a Function with its Derivative)
- Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: Sums, Products, and Powers Of Congruences (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`)
- Corollary: Taylor's Formula for Polynomials (related to Theorem: Taylor's Formula)
- Corollary: The absolute value makes the set of rational numbers a metric space. (related to Definition: Absolute Value of Rational Numbers)
- Corollary: The set of WHILE-computable functions is included in the set of partially WHILE-computable functions (related to Definition: WHILE-Computable Functions)
- Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines)
- Corollary: There is no set of all sets (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Uniqueness of the Empty Set (related to Axiom: Axiom of Empty Set)
- Corollary: Value of Zero to the Power of X (related to Proposition: Limits of General Powers)
- Corollary: `\((-x)(-y)=xy\)` (related to Corollary: `\((-x)y=-(xy)\)`)
- Corollary: `\((-x)y=-(xy)\)` (related to Proposition: Uniqueness of Negative Numbers)
- Corollary: `\((x^{-1})^{-1}=x\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
- Corollary: `\(-(-x)=x\)` (related to Proposition: Uniqueness of Negative Numbers)
- Corollary: `\(-0=0\)` (related to Proposition: Uniqueness of Negative Numbers)
- Corollary: `\(0x=0\)` (related to Proposition: Distributivity Law For Real Numbers)
- Corollary: `\(1^{-1}=1\)` (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
- Definition: "Lies on" Relation
- Definition: (Unit) Ring
- Definition: (Weighted) Arithmetic Mean
- Definition: 1.01: Point
- Definition: 1.02: Line, Curve
- Definition: 1.03: Intersections of Lines
- Definition: 1.04: Straight Line, Segment and Ray
- Definition: 1.05: Surface
- Definition: 1.06: Intersections of Surfaces
- Definition: 1.07: Plane
- Definition: 1.08: Plane Angle
- Definition: 1.09: Angle, Rectilinear, Vertex, Legs
- Definition: 1.10: Right Angle, Perpendicular Straight Lines
- Definition: 1.11: Obtuse Angle
- Definition: 1.12: Acute Angle
- Definition: 1.13: Boundary
- Definition: 1.14: Plane Figure
- Definition: 1.15: Circle, Circumference, Radius
- Definition: 1.16: Center of the Circle
- Definition: 1.17: Diameter of the Circle
- Definition: 1.18: Semicircle
- Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure
- Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle
- Definition: 1.21: Right Triangle, Obtuse Triangle, Acute Triangle
- Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium
- Definition: 1.23: Parallel Straight Lines
- Definition: 10.02: Magnitudes Commensurable and Incommensurable in Square
- Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides
- Definition: 2.2: Gnomon
- Definition: 3.01: Congruent Circles
- Definition: 3.02: Tangent to the Circle, Straight Line Touching The Circle
- Definition: 3.03: Circles Touching One Another
- Definition: 3.04: Chords Equally Far From the Center of a Circle
- Definition: 3.05: Chords Being Further from the Center of a Circle
- Definition: 3.06: Segment of a Circle, Arc
- Definition: 3.07: Angle of a Segment
- Definition: 3.08: Angle in the Segment (Inscribed Angle)
- Definition: 3.09: Angle Standing Upon An Arc
- Definition: 3.10: Circular Sector, Central Angle
- Definition: 3.11: Similar Circular Segments
- Definition: 4.1: Rectilinear Figure Inscribed in Another Rectilinear Figure
- Definition: 4.2: Rectilinear Figure Circumscribed about Another Rectilinear Figure
- Definition: 4.3: Inscribing Rectilinear Figures in Circles
- Definition: 4.4: Circumscribing Rectilinear Figures about Circles
- Definition: 4.5: Inscribing Circles in Rectilinear Figures
- Definition: 4.6: Circumscribing Circles about Rectilinear Figures
- Definition: 4.7: Chord and Secant
- Definition: 5.01: Magnitude is Aliquot Part
- Definition: 5.02: Multiple of a Real Number
- Definition: 5.03: Ratio
- Definition: 5.04: Having a Ratio
- Definition: 5.05: Having the Same Ratio
- Definition: 5.06: Proportional Magnitudes
- Definition: 5.07: Having a Greater Ratio
- Definition: 5.08: Proportion in Three Terms
- Definition: 5.09: Squared Ratio
- Definition: 5.10: Cubed Ratio
- Definition: 5.11: Corresponding Magnitudes
- Definition: 5.12: Alternate Ratio
- Definition: 5.13: Inverse Ratio
- Definition: 5.14: Composition of a Ratio
- Definition: 5.15: Separation of a Ratio
- Definition: 5.16: Conversion of a Ratio
- Definition: 5.17: Ratio ex Aequali
- Definition: 5.18: Perturbed Proportion
- Definition: 6.01: Similar Rectilinear Figures
- Definition: 6.02: Cut in Extreme and Mean Ratio
- Definition: 6.03: Height of a Figure
- Definition: 7.01: Unit
- Definition: 7.02: Number
- Definition: 7.03: Proper Divisor
- Definition: 7.04: Aliquant Part, a Number Being Not a Divisor of Another Number
- Definition: 7.05: Multiple, Number Multiplying another Number
- Definition: 7.06: Even Number
- Definition: 7.07: Odd Number
- Definition: 7.08: Even-Times-Even Number
- Definition: 7.09: Even-Times-Odd Number
- Definition: 7.10: Odd-Times-Odd Number
- Definition: 7.11: Prime Number
- Definition: 7.12: Co-prime (Relatively Prime) Numbers
- Definition: 7.13: Composite Number
- Definition: 7.14: Not Co-prime Numbers
- Definition: 7.15: Multiplication of Numbers
- Definition: 7.16: Rectangular Number, Plane Number
- Definition: 7.17: Cuboidal Number, Solid Number
- Definition: 7.18: Square Number
- Definition: 7.19: Cubic Number, Cube Number
- Definition: 7.20: Proportional Numbers
- Definition: 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers
- Definition: 7.22: Perfect Number
- Definition: Absolute Value of Complex Numbers
- Definition: Absolute Value of Integers
- Definition: Absolute Value of Rational Numbers
- Definition: Absolute Value of Real Numbers (Modulus)
- Definition: Absolute Values and Non-Archimedean Absolute Values of Fields
- Definition: Absolutely Convergent Complex Series
- Definition: Absolutely Convergent Series
- Definition: Abstract Syntax Tree
- Definition: Accumulation Point (Real Numbers)
- Definition: Accumulation Points (Complex Numbers)
- Definition: Addition of Complex Numbers
- Definition: Addition of Ideals
- Definition: Adjacency List Representation
- Definition: Adjacency Matrix
- Definition: Affine Basis, Affine Coordinate System
- Definition: Affine Space
- Definition: Affine Subspace
- Definition: Affinely Dependent and Affinely Independent Points
- Definition: Algebra over a Ring
- Definition: Algebraic Element
- Definition: Algebraic Structure (Algebra)
- Definition: Algorithm (Effective Procedure)
- Definition: Alternating Multilinear Map
- Definition: Altitude of a Triangle
- Definition: Ambiguous and Unambiguous Grammars
- Definition: Argument of a Complex Number
- Definition: Arithmetic Function
- Definition: Associate
- Definition: Associativity
- Definition: Asymptotical Approximation
- Definition: Automorphism
- Definition: Average Velocity
- Definition: Axioms
- Definition: Basis, Coordinate System
- Definition: Bernoulli Experiment
- Definition: Biconnected Graphs, `\(k\)`-Connected Graphs
- Definition: Big O Notation
- Definition: Bijective Function
- Definition: Bilinear Form
- Definition: Binary Operation
- Definition: Binomial Coefficients
- Definition: Bipartite Graph
- Definition: Boolean Algebra
- Definition: Boundary Points, Closures, Interiors, and Exteriors
- Definition: Bounded Affine Set
- Definition: Bounded Complex Sequences
- Definition: Bounded Complex Sets
- Definition: Bounded Real Sequences, Upper and Lower Bounds for a Real Sequence
- Definition: Bounded Sequence
- Definition: Bounded Subset of a Metric Space
- Definition: Bounded Subsets of Ordered Sets
- Definition: Bounded Subsets of Unordered Sets
- Definition: Bounded and Unbounded Functions
- Definition: Cancellation Property
- Definition: Canonical Normal Form
- Definition: Canonical Projection
- Definition: Canonical Representation of Natural Numbers, Factorization
- Definition: Canonical Representation of Positive Rational Numbers
- Definition: Carrier Set
- Definition: Cartesian Product
- Definition: Cauchy Sequence
- Definition: Certain and Impossible Event
- Definition: Characteristic of a Field
- Definition: Characteristic of a Ring
- Definition: Chromatic Number and `$k$`-Coloring of a Graph
- Definition: Closed Curve, Open Curve
- Definition: Closed Walks, Closed Trails, and Cycles
- Definition: Closed and Open Regions of the Complex Plane
- Definition: Closure
- Definition: Co-prime Numbers
- Definition: Coefficient Matrix
- Definition: Collinear Points
- Definition: Collinear Points, Segments, Rays
- Definition: Column Vectors and Row Vectors
- Definition: Combinations
- Definition: Commutative (Abelian) Group
- Definition: Commutative (Unit) Ring
- Definition: Commutativity
- Definition: Comparing the Elements of Posets and Chains
- Definition: Comparison of Cardinal Numbers
- Definition: Comparison of Filters, Finer and Coarser Filters
- Definition: Complement Graph
- Definition: Complete Bipartite Graph
- Definition: Complete Graph
- Definition: Complete Metric Space
- Definition: Complete Ordered Field
- Definition: Complete Residue System
- Definition: Complete System of Representatives
- Definition: Complex Cauchy Sequence
- Definition: Complex Conjugate
- Definition: Complex Infinite Series
- Definition: Complex Polynomials
- Definition: Complex Sequence
- Definition: Composite Number
- Definition: Composition of Binary Relations
- Definition: Computational Problem, Solution
- Definition: Concatenation of Languages
- Definition: Concentric Circles
- Definition: Concurrent Straight Lines
- Definition: Conditional Probability
- Definition: Congruence
- Definition: Congruent, Residue
- Definition: Conjugate Elements of a Group
- Definition: Conjunction
- Definition: Conjunctive and Disjunctive Canonical Normal Forms
- Definition: Connected Vertices
- Definition: Connected and Disconnected Graphs, Bridges and Cutvertices
- Definition: Consistency and Negation-Completeness of a Logical Calculus
- Definition: Constant Function
- Definition: Constant Function Real Case
- Definition: Contained Relation "`$\in_X$`"
- Definition: Continued Fractions
- Definition: Continuous Complex Functions
- Definition: Continuous Function
- Definition: Continuous Functions at Single Complex Numbers
- Definition: Continuous Functions at Single Real Numbers
- Definition: Continuous Functions in Metric Spaces
- Definition: Continuous Random Variables
- Definition: Continuous Real Functions
- Definition: Continuously Differentiable Functions
- Definition: Contrapositive
- Definition: Convergent Complex Sequence
- Definition: Convergent Complex Series
- Definition: Convergent Rational Sequence
- Definition: Convergent Real Sequence
- Definition: Convergent Real Series
- Definition: Convergent Sequences and Limits
- Definition: Convex Affine Set
- Definition: Convex Hull
- Definition: Convex and Concave Functions
- Definition: Coplanar Points and Straight Lines
- Definition: Cosets
- Definition: Cosine of a Real Variable
- Definition: Cotangent Bundle
- Definition: Countable Set, Uncountable Set
- Definition: Curves In the Multidimensional Space `\(\mathbb R^n\)`
- Definition: Cycle Graph
- Definition: Cycles
- Definition: Cyclic Group, Order of an Element
- Definition: Cyclic, Acyclic Graph
- Definition: Decagon
- Definition: Decimal Representation of Real Numbers
- Definition: Def. 10.01: Magnitudes Commensurable and Incommensurable in Length
- Definition: Def. 10.03: Rational and Irrational Magnitudes
- Definition: Def. 10.04: Rational and Irrational Magnitudes in Square
- Definition: Def. 10.05: First Binomial
- Definition: Def. 10.06: Second Binomial
- Definition: Def. 10.07: Third Binomial
- Definition: Def. 10.08: Fourth Binomial
- Definition: Def. 10.09: Fifth Binomial
- Definition: Def. 10.10: Sixth Binomial
- Definition: Def. 10.11: First Apotome
- Definition: Def. 10.12: Second Apotome
- Definition: Def. 10.13: Third Apotome
- Definition: Def. 10.14: Fourth Apotome
- Definition: Def. 10.15: Fifth Apotome
- Definition: Def. 10.16: Sixth Apotome
- Definition: Def. 11.01: Solid Figures, Three-Dimensional Polyhedra
- Definition: Def. 11.02: Surface of a Solid Figure
- Definition: Def. 11.03: Straight Line at Right Angles To a Plane
- Definition: Def. 11.04: Plane at Right Angles to a Plane
- Definition: Def. 11.05: Inclination of a Straight Line to a Plane
- Definition: Def. 11.06: Inclination of a Plane to a Plane
- Definition: Def. 11.07: Similarly Inclined Planes
- Definition: Def. 11.08: Parallel Planes
- Definition: Def. 11.09: Similar Solid Figures
- Definition: Def. 11.10: Equal Solid Figures
- Definition: Def. 11.11: Solid Angle
- Definition: Def. 11.12: Pyramid, Tetrahedron
- Definition: Def. 11.13: Prism, Parallelepiped
- Definition: Def. 11.14: Sphere
- Definition: Def. 11.15: Axis of a Sphere
- Definition: Def. 11.16: Center of a Sphere
- Definition: Def. 11.17: Diameter of a Sphere
- Definition: Def. 11.18: Cone
- Definition: Def. 11.19: Axis of a Cone
- Definition: Def. 11.20: Base of a Cone
- Definition: Def. 11.21: Cylinder
- Definition: Def. 11.22: Axis of a Cylinder
- Definition: Def. 11.23: Bases of a Cylinder
- Definition: Def. 11.24: Similar Cones, Similar Cylinders
- Definition: Def. 11.25: Cube
- Definition: Def. 11.26: Octahedron
- Definition: Def. 11.27: Icosahedron
- Definition: Def. 11.28: Dodecahedron
- Definition: Definition of Complex Numbers
- Definition: Definition of Irrational Numbers
- Definition: Degree Sequence
- Definition: Dense Sets, Nowhere Dense Sets
- Definition: Dependent and Independent Absolute Values
- Definition: Derivability Property
- Definition: Derivative of an n-Dimensional Curve
- Definition: Derivative, Differentiable Functions
- Definition: Derived, Dense-in-itself, and Perfect Sets
- Definition: Describing a Straight Line Using Two Vectors
- Definition: Deterministic Finite Automaton (DFA)
- Definition: Diagonal
- Definition: Diagonal Matrix
- Definition: Diameter In Metric Spaces
- Definition: Difference Operator
- Definition: Difference Quotient
- Definition: Differentiable Manifold, Atlas
- Definition: Differential Form of Degree k
- Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph
- Definition: Dimension of a Vector Space
- Definition: Dimension of an Affine Space
- Definition: Diophantine Equations
- Definition: Direct Product of Groups
- Definition: Direct Sum of Vector Spaces
- Definition: Directional Derivative
- Definition: Discrete Random Variables
- Definition: Discrete and Indiscrete Topology
- Definition: Disjoint Sets
- Definition: Disjunction
- Definition: Divergent Sequences
- Definition: Divergent Series
- Definition: Divisibility of Ideals
- Definition: Division of Real Numbers
- Definition: Divisor, Complementary Divisor, Multiple
- Definition: Divisor-Closed Sets
- Definition: Domain of Discourse
- Definition: Dot Product of Complex Numbers
- Definition: Dot Product, Inner Product, Scalar Product (Complex Case)
- Definition: Dot Product, Inner Product, Scalar Product (General Field Case)
- Definition: Dual Planar Graph
- Definition: Eigenvalue
- Definition: Eigenvector
- Definition: Elementary Gaussian Operations
- Definition: Elementary Symmetric Functions
- Definition: Ellipse
- Definition: Embedding, Inclusion Map
- Definition: Endomorphism
- Definition: Epimorphism
- Definition: Epsilon Non-Deteriministic Finite Automaton (`$\epsilon$`-NFA)
- Definition: Equipotent Sets
- Definition: Equivalence
- Definition: Equivalence Class
- Definition: Equivalence Relation
- Definition: Equivalent Grammars
- Definition: Euclidean Affine Space
- Definition: Euclidean Movement - Isometry
- Definition: Euclidean Ring, Generalization of Division With Quotient and Remainder
- Definition: Euler function
- Definition: Euler's Constant
- Definition: Eulerian Graph
- Definition: Eulerian Tour
- Definition: Even Complex Sequence
- Definition: Even and Odd Complex Functions
- Definition: Even and Odd Functions
- Definition: Even and Odd Numbers
- Definition: Exclusive Disjunction
- Definition: Existence of a Neutral Element
- Definition: Exponential Function of General Base
- Definition: Exponentiation in a Group
- Definition: Exponentiation in a Monoid
- Definition: Extended Real Numbers
- Definition: Extensional Relation
- Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
- Definition: Exterior, Interior, Alternate and Corresponding Angles
- Definition: Face Degree
- Definition: Face, Infinite Face
- Definition: Factorial Polynomials
- Definition: Factorial Ring, Generalization of Factorization
- Definition: Falling And Rising Factorial Powers
- Definition: Falling and Rising Factorial Powers of Functions
- Definition: Field
- Definition: Field Extension
- Definition: Field Homomorphism
- Definition: Finite Field
- Definition: Finite Field Extension
- Definition: Finite Set, Infinite Set
- Definition: Finite and Infinite Graphs
- Definition: Finite and Sigma-Finite Measure
- Definition: Finite and Sigma-Finite Pre-measure
- Definition: First and Second Category Sets
- Definition: First-Order Ordinary Differential Equation (ODE)
- Definition: Fixed Point, Fixed Point Property
- Definition: Floor and Ceiling Functions
- Definition: Formal Languages Generated From a Grammar
- Definition: Frame of Reference
- Definition: Function, Arity and Constant
- Definition: Functional
- Definition: Functional Equation
- Definition: GOTO Command, GOTO Program, Index
- Definition: GOTO-Computable Functions
- Definition: Gaussian Method to Solve Systems of Linear Equations, Rank of a Matrix
- Definition: Generalization of Divisor and Multiple
- Definition: Generalization of the Greatest Common Divisor
- Definition: Generalization of the Least Common Multiple
- Definition: Generalized Polynomial Function
- Definition: Generalized Union of Sets
- Definition: Generating Set of a Group
- Definition: Generating Set of an Ideal
- Definition: Generating Systems
- Definition: Geometric Mean
- Definition: Geometric Probability
- Definition: Geometric Progression, Continued Proportion
- Definition: Girth and Circumference
- Definition: Grammar (Syntax)
- Definition: Graph Decomposable Into `\(k\)` Trees
- Definition: Graph of a Function
- Definition: Group
- Definition: Group Homomorphism
- Definition: Group Operation
- Definition: Group Order
- Definition: Hamiltonian Cycle
- Definition: Hamiltonian Graph
- Definition: Harmonic Series
- Definition: Heine-Borel Property Defines Compact Subsets
- Definition: Hereditary and Weakly Hereditary Properties
- Definition: Hexagon
- Definition: Higher Order Directional Derivative
- Definition: Higher-Order Derivatives
- Definition: Homeomorphism, Homeomorphic Spaces
- Definition: Homomorphism
- Definition: Hyperbolic Cosine
- Definition: Hyperbolic Sine
- Definition: Hyperplane of a Number Space
- Definition: Ideal
- Definition: Identity Function
- Definition: Identity Matrix
- Definition: Implication
- Definition: Improper Integral
- Definition: Incidence
- Definition: Incidence, Adjacency, Neighbours
- Definition: Incidence, Adjacency, Predecessor and Successor Vertices, Neighbours
- Definition: Indefinite Sum, Antidifference
- Definition: Independent Events
- Definition: Index Set and Set Family
- Definition: Indicator (Characteristic) Function, Carrier
- Definition: Inductive Set
- Definition: Inertial and Noninertial Frames of Reference
- Definition: Infimum of Extended Real Numbers
- Definition: Infimum, Greatest Lower Bound
- Definition: Infinite Series, Partial Sums
- Definition: Injective Function
- Definition: Instantaneous Velocity
- Definition: Integral Closure
- Definition: Integral Element
- Definition: Interior, Boundary, and Closures of a Region in the Complex Plane
- Definition: Interlacing Pieces with Respect to a Cycle, Interlacement Graph
- Definition: Interpretation of Propositions - the Law of the Excluded Middle
- Definition: Interpretation of Strings of a Formal Language and Their Truth Function
- Definition: Inverse Element
- Definition: Inverse Relation
- Definition: Invertible Functions, Inverse Functions
- Definition: Invertible and Inverse Matrix
- Definition: Irreducible Polynomial
- Definition: Irreducible, Prime
- Definition: Irreflexive, Asymmetric and Antisymmetric Binary Relations
- Definition: Isolated Point (Real Numbers)
- Definition: Isolated, Adherent, Limit, `$\omega$`-Accumulation and Condensation Points
- Definition: Isometry
- Definition: Isomorphic Digraphs
- Definition: Isomorphic Undirected Graphs
- Definition: Isomorphism
- Definition: Iteration of Languages, Kleene Star, Kleene Plus
- Definition: Jacobi Symbol
- Definition: Jordan Arc (Simple Curve)
- Definition: Knot
- Definition: Knot Diagram, Classical Crossing, Virtual Crossing
- Definition: LOOP Command, LOOP Program
- Definition: LOOP-Computable Functions
- Definition: Language
- Definition: Laplace Experiments and Elementary Events
- Definition: Leaf
- Definition: Legendre Symbol
- Definition: Limit Inferior
- Definition: Limit Ordinal
- Definition: Limit Superior
- Definition: Limit of a Function
- Definition: Limits and Accumulation Points of Sequences
- Definition: Limits of Complex Functions
- Definition: Limits of Real Functions
- Definition: Linear Combination
- Definition: Linear Equations with many Unknowns
- Definition: Linear Function
- Definition: Linear Map
- Definition: Linear Span
- Definition: Linearly Dependent and Linearly Independent Vectors, Zero Vector
- Definition: Linked List, List Nodes
- Definition: Literals, Minterms, and Maxterms
- Definition: Local Extremum
- Definition: Logarithmically Convex and Concave Functions
- Definition: Logical Arguments
- Definition: Logical Calculus
- Definition: Magma
- Definition: Manifold
- Definition: Matrix Multiplication
- Definition: Matrix and Vector Addition
- Definition: Matrix, Set of Matrices over a Field
- Definition: Maximal Ideal
- Definition: Maximum (Real Numbers)
- Definition: Measurable Set
- Definition: Measurable Space
- Definition: Measure
- Definition: Measureable Function
- Definition: Meter
- Definition: Metric (Distance)
- Definition: Metric Space
- Definition: Minimal Inductive Set
- Definition: Minimal Polynomial
- Definition: Minimal Tree Decomposability
- Definition: Minimum (Real Numbers)
- Definition: Module
- Definition: Modulo Operation for Real Numbers
- Definition: Modulus of Continuity of a Continuous Function
- Definition: Monoid
- Definition: Monomorphism
- Definition: Monotonic Functions
- Definition: Monotonic Sequences
- Definition: Mostowski Function and Collapse
- Definition: Multilinear Map
- Definition: Multiplication of Complex Numbers
- Definition: Multiplication of Natural Numbers
- Definition: Multiplicative Functions
- Definition: Multiplicative System
- Definition: Multiplicity of a Root of a Polynomial
- Definition: Mutually Disjoint Sets
- Definition: Mutually Exclusive and Collectively Exhaustive Events
- Definition: Mutually Independent Events
- Definition: Möbius Function, Square-free
- Definition: Negation
- Definition: Negation of a String
- Definition: Neighborhood
- Definition: Nested Real Intervals
- Definition: Non-deterministic Finite Automaton (NFA)
- Definition: Norm, Normed Vector Space
- Definition: Normal Subgroups
- Definition: Null Graph
- Definition: Number `$\pi$`
- Definition: Number of Divisors
- Definition: Odd Complex Sequence
- Definition: One-sided Derivative, Right-Differentiability and Left-Differentiability
- Definition: Open Ball, Neighborhood
- Definition: Open Cover
- Definition: Open Function, Closed Function
- Definition: Open Sets in Metric Spaces
- Definition: Open and Closed Discs
- Definition: Open and Closed Functions
- Definition: Open, Closed, Clopen
- Definition: Order Embedding
- Definition: Order Relation for Integers - Positive and Negative Integers
- Definition: Order Relation for Natural Numbers
- Definition: Order Relation for Rational Numbers - Positive and Negative Rational Numbers
- Definition: Order Relation for Step Functions
- Definition: Order Relation of Real Numbers
- Definition: Order of a Graph
- Definition: Ordered Field
- Definition: Ordered Pair, n-Tuple
- Definition: Ordering of Topologies
- Definition: Ordinal Number
- Definition: Pairwise Independent Events
- Definition: Paradox
- Definition: Parallelogram - Defining Property III
- Definition: Partial and Total Maps (Functions)
- Definition: Pentagon
- Definition: Perfect Number
- Definition: Perfect Square
- Definition: Periodic Functions
- Definition: Permutations
- Definition: Perspectivities
- Definition: Pieces of a Graph With Respect to A Cycle
- Definition: Planar Drawing (Embedding)
- Definition: Planar Graph
- Definition: Point of Division, Point of External Division
- Definition: Points in a Coordinate System - Number Spaces
- Definition: Points vs. Vectors in a Number Space
- Definition: Points, Lines, Planes, Hyperplanes
- Definition: Points, Straight Lines, and Planes
- Definition: Pointwise and Uniform Convergence
- Definition: Pointwise and Uniformly Convergent Sequences of Functions
- Definition: Polynomial Ring
- Definition: Polynomial over a Ring, Degree, Variable
- Definition: Polynomials
- Definition: Positive and Negative Parts of a Real-Valued Function
- Definition: Power Set
- Definition: Pre-measure
- Definition: Predicate of a Logical Calculus
- Definition: Preorder, Partial Order and Poset
- Definition: Prime Field
- Definition: Prime Ideal
- Definition: Prime Numbers
- Definition: Prime-Counting Function
- Definition: Principal Ideal
- Definition: Principal Ideal Domain
- Definition: Principal Ideal Ring
- Definition: Probability Distribution
- Definition: Probability Mass Function
- Definition: Probability and its Axioms
- Definition: Products
- Definition: Projectivities, Ranges and Pencils
- Definition: Proofs and Theorems in a Logical Calculus
- Definition: Quadratic Residue, Quadratic Nonresidue
- Definition: Quantifier, Bound Variables, Free Variables
- Definition: Quotient Set, Partition
- Definition: Random Experiments and Random Events
- Definition: Random Variable, Realization, Population and Sample
- Definition: Ratio of Two Real Numbers
- Definition: Rational Cauchy Sequence
- Definition: Rational Functions
- Definition: Rational Sequence
- Definition: Real Absolute Value Function
- Definition: Real Cauchy Sequence
- Definition: Real Identity Function
- Definition: Real Intervals
- Definition: Real Sequence
- Definition: Real Subsequence
- Definition: Rearrangement of Infinite Series
- Definition: Reciprocal Function
- Definition: Recursive Definition of the Determinant
- Definition: Reduced Residue System
- Definition: Reduction of an Integer Polynomial Modulo a Prime Number
- Definition: Reflexive, Symmetric and Transitive Binary Relations
- Definition: Regular Graph
- Definition: Regular Open, Regular Closed
- Definition: Reidemeister Moves, Planar Isotopy Moves, Diagrammatic Moves
- Definition: Relation
- Definition: Relative and Absolute Frequency
- Definition: Restriction
- Definition: Riemann Sum With Respect to a Partition
- Definition: Riemann-Integrable Functions
- Definition: Ring Homomorphism
- Definition: Ring of Integers
- Definition: Ring of Sets (measure-theoretic definition)
- Definition: Root, Degree of a Tree, Subtree, Height
- Definition: Rules of Inference
- Definition: Satisfaction Relation, Model, Tautology, Contradiction
- Definition: Second
- Definition: Section over a Base Space
- Definition: Semantics of PL0
- Definition: Semantics of a Formal Language
- Definition: Semi-Eulerian Graph
- Definition: Semi-Eulerian Tour, Open Trail
- Definition: Semi-Hamiltonian Graph
- Definition: Semi-Hamiltonian Path
- Definition: Semigroup
- Definition: Separating and Non-Separating Cycles
- Definition: Sequence
- Definition: Sequences Tending To Infinity
- Definition: Set Complement
- Definition: Set Difference
- Definition: Set Intersection
- Definition: Set Partition
- Definition: Set Union
- Definition: Set of Natural Numbers (Peano)
- Definition: Set of Truth Values (True and False)
- Definition: Set, Set Element, Empty Set
- Definition: Set-theoretic Definition of Order Relation for Natural Numbers
- Definition: Set-theoretic Definitions of Natural Numbers
- Definition: Sets of Integers Co-Prime To a Given Integer
- Definition: Sieve of Eratosthenes
- Definition: Sieve, Sieve Problem
- Definition: Sigma-Algebra
- Definition: Signature
- Definition: Signature of Propositional Logic - PL0
- Definition: Signum Function in An Ordered Field
- Definition: Similarity
- Definition: Simplex
- Definition: Sine of a Real Variable
- Definition: Singleton
- Definition: Size of a Graph
- Definition: Solution of Ordinary DE
- Definition: Solution to a Lower Triangular SLE - Forward Substitution
- Definition: Solution to an Upper Triangular SLE - Backward Substitution
- Definition: Soundness and Completeness of a Logical Calculus
- Definition: Spacetime Diagram
- Definition: Spanning Subgraph
- Definition: Spanning Tree
- Definition: Special Elements of Ordered Sets
- Definition: Spectrum of a Commutative Ring
- Definition: Square Matrix
- Definition: Step Functions
- Definition: Stirling Numbers of First and Second Kind
- Definition: Strict Total Order, Strictly-ordered Set
- Definition: Strings (words) over an Alphabet
- Definition: Subadditive Function
- Definition: Subbasis and Basis of Topology
- Definition: Subdigraphs and Superdigraphs; Induced Subdigraph
- Definition: Subdivision of a Graph
- Definition: Subfield
- Definition: Subgraphs and Supergraphs; Induced Subgraph
- Definition: Subgroup
- Definition: Subring
- Definition: Subsequence
- Definition: Subset and Superset
- Definition: Subsets of Prime Numbers Not Dividing a Natural Number
- Definition: Subspace
- Definition: Substructure
- Definition: Subtraction of Complex Numbers
- Definition: Subtraction of Integers
- Definition: Subtraction of Rational Numbers
- Definition: Subtraction of Real Numbers
- Definition: Sum of Angles
- Definition: Sum of Divisors
- Definition: Sums
- Definition: Supplemental Angles
- Definition: Suppressing Vertices, Suppressed Multigraph
- Definition: Supremum Norm for Functions
- Definition: Supremum of Extended Real Numbers
- Definition: Supremum, Least Upper Bound
- Definition: Surjective Function
- Definition: Symmetric Bilinear Form
- Definition: Symmetric Matrix
- Definition: Syntax of PL0 - Propositions as Boolean Terms
- Definition: Systems of Linear Equations with many Unknowns
- Definition: Tangent Bundle
- Definition: Tangent of a Real Variable
- Definition: The Class of all Ordinals `$\Omega$`
- Definition: Topological Chart
- Definition: Topological Product, Product Topology
- Definition: Topological Space, Topology
- Definition: Topological Subspaces and Subspace Topologies
- Definition: Topological Sum, Disjoint Union
- Definition: Topological, Continuous, Open, and Closed Invariants
- Definition: Total Order and Chain
- Definition: Total and Unique Binary Relations
- Definition: Totally Differentiable Functions, Total Derivative
- Definition: Transcendental Element
- Definition: Transition Map
- Definition: Transitive Set
- Definition: Transposed Matrix
- Definition: Trees and Forests
- Definition: Triangle
- Definition: Triangle Numbers
- Definition: Truth Table
- Definition: Twin Prime Numbers
- Definition: Type-0 (Phrase Structure) Grammars and Recursively Enumerable Languages
- Definition: Type-1 (cs) Grammars and Context-sensitive Languages
- Definition: Type-2 (cf) Grammars and Context-free Languages
- Definition: Type-3 (Linear) Grammars and Regular Languages
- Definition: Ultrafilter
- Definition: Undirected Graph, Vertices, Edges, Simple Graph
- Definition: Uniformly Continuous Functions (General Metric Spaces Case)
- Definition: Uniformly Continuous Functions (Real Case)
- Definition: Unit
- Definition: Unit-Cost Random Access Machine
- Definition: Unitary Affine Space
- Definition: Universal Set
- Definition: Unknot
- Definition: Upper and Lower Triangular Matrix
- Definition: Variable in a Logical Calculus
- Definition: Vector Field
- Definition: Vector Space
- Definition: Vertex Degrees for Digraphs
- Definition: Vertex Degrees for Undirected Graphs
- Definition: Von Mangoldt Function
- Definition: WHILE Command, WHILE Program
- Definition: WHILE-Computable Functions
- Definition: Walks, Trails, and Paths
- Definition: Weakly and Strongly Connected Digraphs
- Definition: Well-founded Relation
- Definition: Well-order, Well-ordered Set
- Definition: Zariski Topology of a Commutative Ring
- Definition: Zero Divisor and Integral Domain
- Definition: Zero Matrix, Zero Vector
- Definition: Zero Ring
- Definition: Zero of a Function
- Definition: `$\mathcal P$`-Computable and `$\mathcal P$`-Decidable Problems
- Definition: `$k$`-nary Connectives, Prime and Compound Propositions
- Definition: `\(C^n\)` Differentiable Function
- Definition: `\(C^{n}\)`-Diffeomorphism
- Definition: `\(b\)`-Adic Fractions
- Definition: `\(n\)` times Continuously Differentiable Functions
- Definition: n-Periodical Complex Sequence
- Epoch: 11th Century
- Epoch: 12th Century
- Epoch: 13th Century
- Epoch: 14th Century
- Epoch: 15th Century
- Epoch: 16th Century
- Epoch: 17th Century
- Epoch: 18th Century
- Epoch: 19th Century
- Epoch: 20th Century
- Epoch: 21th Century
- Epoch: Ancient World
- Epoch: Early Middle Ages
- Epoch: Late Ancient World
- Epoch: Prehistory
- Example: A digraph with a loop, and parallel and inverse edges (related to Definition: Digraph, Initial and Terminal Vertices, Loops, Parallel and Inverse Edges, Simple Digraph)
- Example: A pointwise convergent sequence of functions, which is not uniformly convergent (related to Definition: Pointwise and Uniform Convergence)
- Example: Analytical Thinking Joke (related to Part: Jokes)
- Example: Applications of the Cauchy Condensation Criterion (related to Proposition: Cauchy Condensation Criterion)
- Example: Bad Luck For People Suffering From Triskaidekaphobia (related to Part: Oddities and Curiosities)
- Example: Difference Operators of Some Functions (related to Part: Discrete Calculus and Difference Equations)
- Example: Divergent Alternating Sequence (related to Definition: Divergent Sequences)
- Example: Divisibility Examples (related to Definition: Divisor, Complementary Divisor, Multiple)
- Example: Example of Continuous Functions in Topological Spaces (related to Chapter: Continuity)
- Example: Examples of ASTs (related to Part: Formal Languages)
- Example: Examples of Absolute Values (related to Part: Ordered Fields and Their Topology)
- Example: Examples of Accumulation Points (related to Definition: Accumulation Point (Real Numbers))
- Example: Examples of Adjacency Matrices (related to Definition: Adjacency Matrix)
- Example: Examples of Boolean Functions (related to Lemma: Boolean Function)
- Example: Examples of Canonical Normal Forms (related to Chapter: Normal Forms in `$PL0$`)
- Example: Examples of Computational Problems (related to Part: Computability)
- Example: Examples of Convergent Sequences in Topological Spaces (related to Chapter: Sequences and Limits)
- Example: Examples of Correct and Incorrect Negations of Implications (related to Lemma: Negation of an Implication)
- Example: Examples of Cyclic Groups (related to Part: Group Theory)
- Example: Examples of DFA (related to Chapter: Finite Automata (Finite Sequential Machines))
- Example: Examples of Equivalence Relations (related to Definition: Equivalence Relation)
- Example: Examples of Fields (related to Chapter: Fields (Overview))
- Example: Examples of Functions Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
- Example: Examples of Functions Not Continuous at a Single Point (related to Definition: Continuous Functions at Single Real Numbers)
- Example: Examples of Functions in a Logical Calculus (related to Definition: Function, Arity and Constant)
- Example: Examples of Group Homomorphisms (related to Definition: Group Homomorphism)
- Example: Examples of Groups (related to Chapter: Groups (Overview))
- Example: Examples of Important Arithmetic Functions (related to Chapter: Arithmetic Functions)
- Example: Examples of Kernels and Images of Group Homomorphisms (related to Definition: Group Homomorphism)
- Example: Examples of Languages (related to Definition: Language)
- Example: Examples of Logical Arguments (related to Part: Methods of Mathematical Proving)
- Example: Examples of Magmas, Semigroups, and Monoids (related to Chapter: Magmas, Semigroups, Monoids (Overview))
- Example: Examples of Multinomial Coefficients (related to Proposition: Multinomial Coefficient)
- Example: Examples of Multiplicative Functions (related to Definition: Multiplicative Functions)
- Example: Examples of NFA (related to Chapter: Finite Automata (Finite Sequential Machines))
- Example: Examples of Paradoxes (related to Definition: Paradox)
- Example: Examples of Predicates in a Logical Calculus (related to Definition: Predicate of a Logical Calculus)
- Example: Examples of Properties of Group Homomorphisms (related to Definition: Group Homomorphism)
- Example: Examples of Propositions With a More Complex Syntax (related to Definition: Interpretation of Propositions - the Law of the Excluded Middle)
- Example: Examples of Quantifiers in a Logical Calculus (related to Definition: Quantifier, Bound Variables, Free Variables)
- Example: Examples of Real Functions, Whose Graphs Cannot be Plotted (related to Chapter: Types of Real Functions)
- Example: Examples of Relations (related to Definition: Relation)
- Example: Examples of Ring Homomorphisms (related to Chapter: Rings (Overview))
- Example: Examples of Strings over Alphabets (related to Definition: Strings (words) over an Alphabet)
- Example: Examples of Syntax (related to Definition: Grammar (Syntax))
- Example: Examples of Variables in a Logical Calculus (related to Definition: Variable in a Logical Calculus)
- Example: Examples of Vector Spaces (related to Definition: Vector Space)
- Example: Examples of Well-founded Relations (related to Definition: Well-founded Relation)
- Example: Examples of `$\epsilon$`-NFA (related to Chapter: Finite Automata (Finite Sequential Machines))
- Example: Existence and Uniqueness of Solutions (related to Part: Methods of Mathematical Proving)
- Example: Existence of not Riemann-Integrable Functions (related to Definition: Riemann-Integrable Functions)
- Example: Fair Dice (related to Definition: Laplace Experiments and Elementary Events)
- Example: Graph of a Real-Valued Function with One Variable (related to Definition: Graph of a Function)
- Example: Is Reality Real? (related to Part: Jokes)
- Example: Mentalist Trick (1) (related to Part: Tricks for Mental Maths)
- Example: Mentalist Trick (2) (related to Part: Tricks for Mental Maths)
- Example: Multiplying small numbers by 9 (related to Part: Tricks for Mental Maths)
- Example: Pizza Lovers (related to Part: Jokes)
- Example: Pointwise vs. Uniformly Convergent Sequences of Functions (related to Section: Uniform Convergence of Functions)
- Example: Sexy Parabola (Joke) (related to Part: Jokes)
- Example: Simulating More Complex Commands Using Basic `\(L O O P\)` Commands (related to Definition: LOOP Command, LOOP Program)
- Example: Solution to a Degenerated Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
- Example: Solution to a Diagonal SLE (related to Section: Solving Simple Systems of Linear Equations)
- Example: Solution to a Zero SLE (related to Section: Solving Simple Systems of Linear Equations)
- Example: Some Other Examples Of Mostowski Functions and Collapses (related to Definition: Mostowski Function and Collapse)
- Example: The Gaussian Method in Practice (related to Section: Solving General Systems Of Linear Equations - Gaussian Method)
- Example: The Paper Box (related to Section: Various Dissection Puzzles)
- Example: Trivial Subspaces, Zero Space (related to Definition: Subspace)
- Example: What is time? (related to Part: Jokes)
- Example: Working with Mostowski Functions and Collapses (related to Definition: Mostowski Function and Collapse)
- Explanation: 1.1: Equality is an Equivalence Relation (related to Subsection: Common Notions (all Books))
- Explanation: 1.2: Adding Equations Preserves Equality (related to Subsection: Common Notions (all Books))
- Explanation: 1.3: Subtracting Equations Preserves Equality (related to Subsection: Common Notions (all Books))
- Explanation: 1.4: Congruent Figures (related to Subsection: Common Notions (all Books))
- Explanation: 1.5: Comparing the Size of Sets and Their Subsets (related to Subsection: Common Notions (all Books))
- Explanation: A Note on Well-ordered Sets (related to Definition: Well-order, Well-ordered Set)
- Explanation: All Types of Morphisms and Their Properties (related to Motivation: Common Concepts Of Algebra: Substructures and Morphisms)
- Explanation: Building the Successors of Ordinal Numbers (related to Part: Ordinal Numbers)
- Explanation: Calculation Rules in a Group with Additive Notation (related to Motivation: Calculations in a Group)
- Explanation: Changing the Index Of Sums in Number Theory (related to Lemma: Möbius and Floor Functions Combined)
- Explanation: Characteristic Zero Instead of Characteristic Infinite (related to Definition: Characteristic of a Ring)
- Explanation: Chomsky's Hierarchy of Languages (related to Chapter: Classification of Formal Languages)
- Explanation: Combinatorial Interpretation of Stirling Numbers of the First Kind (related to Part: Stirling Numbers)
- Explanation: Combinatorial Interpretation of Stirling Numbers of the Second Kind (related to Part: Stirling Numbers)
- Explanation: Comparison Between the Number Systems (related to Branch: Number Systems and Arithmetics)
- Explanation: Connection Between the Cartesian Product and Functions Mapping Finite Sets into a Non-empty Set (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
- Explanation: Connection Between the Power Set and Functions Mapping Sets into a Set with Two Elements (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
- Explanation: Correct Negation of Statements (related to Part: Methods of Mathematical Proving)
- Explanation: Deductive Reasoning (related to Chapter: Logical Arguments Used in Mathematical Proofs)
- Explanation: Einstein's Interpretation of the Time Dilation (related to Proposition: Time Dilation, Lorentz Factor)
- Explanation: Examples of Dual Statements for Planar Graphs (related to Chapter: Conditions for Planarity and Planarity Testing)
- Explanation: Examples of Extensional Relations (related to Definition: Extensional Relation)
- Explanation: Explanation of Congruence Classes (related to Definition: Congruent, Residue)
- Explanation: Explanation of the Heine-Borel Property (related to Definition: Heine-Borel Property Defines Compact Subsets)
- Explanation: Fourteen Sets Formed By Closure, Interior and Complement Operations (related to Chapter: Basic Topological Concepts)
- Explanation: Geometrical Interpretation of Hyperplanes (related to Definition: Hyperplane of a Number Space)
- Explanation: Good Practices for Writing Mathematical Proofs (related to Part: Methods of Mathematical Proving)
- Explanation: Hasse Diagram (related to Chapter: Order Relations)
- Explanation: How Functions interact with Set Operations (related to Lemma: Behavior of Functions with Set Operations)
- Explanation: How a line is different from a solid and a surface? (related to Definition: 1.02: Line, Curve)
- Explanation: How a point is different from a solid, a surface and a line? (related to Definition: 1.01: Point)
- Explanation: How the Axiom of Separation Avoids Russel's Paradox (related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))
- Explanation: How to Interpret Events Which Are Constructed From Other Events Doing Set Operations? (related to Definition: Random Experiments and Random Events)
- Explanation: Indicator Function and Complete Induction (related to Part: Set-theoretic Prerequisites Needed For Combinatorics)
- Explanation: Inductive Reasoning (related to Chapter: Invalid Logical Arguments)
- Explanation: Jacobi Symbol vs. Solvability of Quadratic Congruences (related to Section: Generalizations of the Legendre symbol - Jacobi and Kronecker Symbols)
- Explanation: Kernel and Image (related to Definition: Linear Map)
- Explanation: Local Extrema on a Closed Interval (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
- Explanation: Matrices of Linear Maps (related to Definition: Linear Map)
- Explanation: Mutually Exclusive vs. Pairwise Independent Events (related to Definition: Pairwise Independent Events)
- Explanation: Notes on Special Elements of Posets (related to Chapter: Order Relations)
- Explanation: Operation Table (related to Part: Algebraic Structures - Overview)
- Explanation: Pascal's Triangle (Triangle of Binomial Coefficients) (related to Definition: Combinations)
- Explanation: Possibilities to Describe Sets, Venn-Diagrams, List, and Set-Builder Notations (related to Definition: Set, Set Element, Empty Set)
- Explanation: Properties of the Pascal's Triangle (related to Explanation: Pascal's Triangle (Triangle of Binomial Coefficients))
- Explanation: Representations of Binary Relations (related to Chapter: Binary Relations and Their Properties)
- Explanation: Some Remarks on Functions (related to Chapter: Functions (Maps))
- Explanation: Summary of Different Order Relations (related to Chapter: Order Relations)
- Explanation: Transitive Set and Countability - Natural Numbers Have the Smallest Infinite Cardinality (related to Part: Cardinal Numbers)
- Explanation: Unions of Subgroups Are Not Subgroups (related to Chapter: Groups (Overview))
- Explanation: What does WLOG mean? (related to Part: Methods of Mathematical Proving)
- Explanation: What does the Extensionality Principle mean? (related to Axiom: Axiom of Extensionality)
- Explanation: When is something "well-defined" in mathematics? (related to Part: Methods of Mathematical Proving)
- Explanation: Why did Euclid postulate the axiom of straight line determined by two points? (related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
- Explanation: Why do the Peano axioms define natural numbers? (related to Axiom: Peano Axioms)
- Explanation: Why is a random variable neither random, nor variable? (related to Definition: Random Variable, Realization, Population and Sample)
- Explanation: Why is it impossible to divide by `\(0\)`? (related to Proposition: Definition of Real Numbers)
- Index
- Index: By Building Blocks
- Index: By Contributors. Thank you!
- Index: By Site Issues
- Index: Content Tree
- Index: Keyword
- Index: Person
- Index: Symbolic Notation
- Index: Widgets
- Lemma: A Criterion for Associates
- Lemma: A Criterion for Valid Logical Arguments
- Lemma: A proposition cannot be both, true and false
- Lemma: A proposition cannot be equivalent to its negation
- Lemma: Abel's Lemma for Testing Convergence
- Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals
- Lemma: Affirming the Consequent of an Implication
- Lemma: Any Positive Characteristic Is a Prime Number
- Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull
- Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions
- Lemma: Behavior of Functions with Set Operations
- Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph
- Lemma: Boolean Algebra of Propositional Logic
- Lemma: Boolean Function
- Lemma: Characterization of Closed Sets by Limits of Sequences
- Lemma: Coloring of Trees
- Lemma: Comparing the Elements of Strictly Ordered Sets
- Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis
- Lemma: Composition of Functions
- Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property
- Lemma: Composition of Relations Preserves Their Right-Uniqueness Property
- Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms
- Lemma: Continuants and Convergents
- Lemma: Convergence Test for Telescoping Series
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are Rational Cauchy Sequences
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are a Subgroup of Rational Cauchy Sequences With Respect To Addition
- Lemma: Convergent Rational Sequences With Limit `\(0\)` Are an Ideal Of the Ring of Rational Cauchy Sequences
- Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces)
- Lemma: Coprimality and Congruence Classes
- Lemma: Criteria for Convergent Sequences
- Lemma: Cyclic Groups are Abelian
- Lemma: De Morgan's Laws (Logic)
- Lemma: Decreasing Sequence of Suprema of Extended Real Numbers
- Lemma: Denying the Antecedent of an Implication
- Lemma: Disjunctive Syllogism
- Lemma: Distributivity of Conjunction and Disjunction
- Lemma: Divisibility of Principal Ideals
- Lemma: Division with Quotient and Remainder
- Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree
- Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE
- Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors
- Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals
- Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace
- Lemma: Euler's Identity
- Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition
- Lemma: Every Proposition Implies Itself
- Lemma: Factor Groups
- Lemma: Factor Rings, Generalization of Congruence Classes
- Lemma: Fiber of Maximal Ideals
- Lemma: Fiber of Prime Ideals
- Lemma: Fiber of Prime Ideals Under a Spectrum Function
- Lemma: Finite Cardinal Numbers and Set Operations
- Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point
- Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations
- Lemma: Gaussian Lemma (Number Theory)
- Lemma: Generalized Euclidean Lemma
- Lemma: Greatest Common Divisor and Least Common Multiple of Ideals
- Lemma: Group Homomorphisms and Normal Subgroups
- Lemma: Handshaking Lemma for Finite Digraphs
- Lemma: Handshaking Lemma for Finite Graphs
- Lemma: Handshaking Lemma for Planar Graphs
- Lemma: Hypothetical Syllogism
- Lemma: Implication as a Disjunction
- Lemma: Increasing Sequence of Infima of Extended Real Numbers
- Lemma: Invertible Functions on Real Intervals
- Lemma: It is true that something can be (either) true or false
- Lemma: Kernel and Image of Group Homomorphism
- Lemma: Kernel and Image of a Group Homomorphism are Subgroups
- Lemma: LOOP-Computable Functions are Total
- Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Lemma: Lem. 10.021: Medial is Irrational
- Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square
- Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square
- Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II
- Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational
- Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
- Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them
- Lemma: Lem. 10.13: Finding Pythagorean Magnitudes
- Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares
- Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters
- Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere
- Lemma: Lem. 13.18: Angle of the Pentagon
- Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`
- Lemma: Lower Bound of Leaves in a Tree
- Lemma: Mixing-up the Inclusive and Exclusive Disjunction
- Lemma: Mixing-up the Sufficient and Necessary Conditions
- Lemma: Modus Ponens
- Lemma: Modus Tollens
- Lemma: Möbius and Floor Functions Combined
- Lemma: Negation of an Implication
- Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring
- Lemma: Prime Ideals of Multiplicative Systems in Integral Domains
- Lemma: Properties of Ordinal Numbers
- Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition
- Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication
- Lemma: Reciprocity Law for Floor Functions
- Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree
- Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine
- Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed
- Lemma: Sieve for Twin Primes
- Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices
- Lemma: Splitting a Graph with Even Degree Vertices into Cycles
- Lemma: Stirling Numbers and Rising Factorial Powers
- Lemma: Subgroups and Their Cosets are Equipotent
- Lemma: Subgroups of Cyclic Groups
- Lemma: Successor of Ordinal
- Lemma: Sum of Möbius Function Over Divisors With Division
- Lemma: Sum of Roots Of Unity in Complete Residue Systems
- Lemma: Sums of Floors
- Lemma: The Proving Principle By Contraposition, Contrapositive
- Lemma: The Proving Principle by Complete Induction
- Lemma: The Proving Principle by Contradiction
- Lemma: The Proving Principle by Transfinite Induction
- Lemma: Trapezoid Rule
- Lemma: Unique Valuation of Minterms and Maxterms
- Lemma: Uniqueness Lemma of a Finite Basis
- Lemma: Unit Circle
- Lemma: Unit Ring of All Rational Cauchy Sequences
- Lemma: Upper Bound for the Product of General Powers
- Lemma: Upper Bound of Harmonic Series Times Möbius Function
- Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?
- Lemma: Zorn's Lemma
- Motivation: Burali-Forti Paradox (related to Part: Ordinal Numbers)
- Motivation: Calculations in a Group (related to Chapter: Groups (Overview))
- Motivation: Cantor's Astonishing Discoveries Regarding the Cardinals of Infinite Sets (related to Part: Cardinal Numbers)
- Motivation: Common Concepts Of Algebra: Substructures and Morphisms (related to Part: Algebraic Structures - Overview)
- Motivation: Formulae of Negative Factorial Powers Explained (related to Definition: Falling And Rising Factorial Powers)
- Motivation: Is "Being a Set Element" ("`$\in$`") a Relation? (related to Part: Ordinal Numbers)
- Motivation: Measurement of the Speed of Light - First Experimental Proof that the Speed of Light is Finite (related to Part: Special Relativity)
- Motivation: Michelson-Morley Experiment (related to Part: Special Relativity)
- Motivation: Motivation for the Proof of Uniqueness of Zero (related to Proposition: Uniqueness of Negative Numbers)
- Motivation: Observation 1: The Mostowski Function Produces Transitive Sets (related to Definition: Mostowski Function and Collapse)
- Motivation: Observation 2: The Mostowski Function (Sometimes) Produces Relation Embeddings (related to Definition: Mostowski Function and Collapse)
- Motivation: Rational Numbers and the Greatest Common Divisor (related to Chapter: Algebraic and Transcendent Numbers)
- Motivation: Russell's Paradox (related to Branch: Set Theory)
- Motivation: Significance of Equivalence Relations (related to Definition: Equivalence Relation)
- Motivation: Usage of Ordered Tuples In Other Mathematical Disciplines (related to Part: Relations)
- Motivation: What does not have to be proved in mathematics? (related to Part: Methods of Mathematical Proving)
- Part: Additive Number Theory
- Part: Algebraic Number Theory (Link)
- Part: Algebraic Number Theory and Ring Theory
- Part: Algebraic Structures - Overview
- Part: Algebraic Topology
- Part: Analytic Geometry
- Part: Analytic Number Theory
- Part: Basic Algorithms
- Part: Basic Concepts of Logic
- Part: Basics Concepts in Graph Theory
- Part: Basics about Sets
- Part: Calculus of Variations
- Part: Cardinal Numbers
- Part: Classical Physics
- Part: Complex Analysis
- Part: Complex Numbers
- Part: Computability
- Part: Computational Complexity Theory
- Part: Constructions with Ruler and Compass
- Part: Continuous Distributions
- Part: Cycles, Permutations, Combinations and Variations
- Part: Data Structures
- Part: Differential Equations
- Part: Differential Geometry
- Part: Discrete Calculus and Difference Equations
- Part: Discrete Distributions
- Part: Dudeney's Amusements in Mathematics
- Part: Elementary Number Theory
- Part: Elliptic Geometry
- Part: Estimation Theory
- Part: Euclidean Geometry
- Part: Extremal Problems in Graph Theory
- Part: Finite Fields
- Part: Flows
- Part: Forgotten Reckoning Methods
- Part: Formal Languages
- Part: Functional Analysis
- Part: Galois Theory
- Part: General Relativity
- Part: Graph Transformations
- Part: Group Theory
- Part: Gödel's Incompleteness Theorems
- Part: Higher-Order Logics
- Part: Historical Development of Analysis
- Part: Historical Development of Combinatorics
- Part: Historical Development of Geometry
- Part: Historical Development of Logic
- Part: Historical Development of Number Theory
- Part: Historical Development of Set Theory
- Part: Historical Development of Topology
- Part: Homotopy
- Part: Hyperbolic Geometry
- Part: Infinite Graphs
- Part: Integers
- Part: Irrational Numbers
- Part: Jokes
- Part: Laws of Large Numbers
- Part: Linear Algebra
- Part: Linear Integral Equations
- Part: Markov Chains
- Part: Matchings
- Part: Methods of Mathematical Proving
- Part: Metric Spaces
- Part: Multidimensional Random Variables
- Part: Natural Numbers
- Part: Network Design and Routing
- Part: Numerical Algorithms
- Part: Oddities and Curiosities
- Part: Optimization Methods
- Part: Ordered Fields and Their Topology
- Part: Ordinal Numbers
- Part: PL1 - First Order Predicate Logic
- Part: Paths, Cycles and Connectivity
- Part: Planar Graphs
- Part: Projective Geometry
- Part: Proof Theory
- Part: Propositional Logic
- Part: Quantum Field Theory
- Part: Quantum Mechanics
- Part: Randomness and Probability Calculus
- Part: Rational Numbers
- Part: Real Analysis of Multiple Variables
- Part: Real Analysis of One Variable and Elements of Complex Analysis
- Part: Real Numbers
- Part: Relations
- Part: Semi-numerical Algorithms
- Part: Separation Of Topological Spaces
- Part: Set-theoretic Prerequisites Needed For Combinatorics
- Part: Solving Strategies and Sample Solutions Related to Arithmetics
- Part: Solving Strategies and Sample Solutions Related to Number Theory
- Part: Solving Strategies and Sample Solutions to Problems in Algebra
- Part: Solving Strategies and Sample Solutions to Problems in Combinatorics
- Part: Solving Strategies and Sample Solutions to Problems in Graph Theory
- Part: Solving Strategies and Sample Solutions to Problems in Logic
- Part: Solving Strategies and Sample Solutions to Problems in Set Theory
- Part: Some Unsolved Number-Theoretic Problems
- Part: Special Relativity
- Part: Stirling Numbers
- Part: The Basics Set-theoretic Topology
- Part: Transitive Hull and Irreducible Kernels
- Part: Tricks for Mental Maths
- Part: Vector Analysis
- Part: Vertex Colorings and Decompositions
- Part: Zermelo-Fraenkel Set Theory
- Person: 'sGravesande, Willem Jacob
- Person: Aaboe, Asger Hartvig
- Person: Abbe, Ernst
- Person: Abbott, Edwin
- Person: Abel, Niels Henrik
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- Person: Abhyankar, Shreeram Shankar
- Person: Abraham Ibn Ezra, ben Meir
- Person: Abraham, Max
- Person: Abramescu, Nicolae
- Person: Abu Kamil, ibn Aslam Shuja
- Person: Ackermann, Wilhelm
- Person: Adam, Pedro Puig
- Person: Adams (2), Edwin P.
- Person: Adams (3), Doris
- Person: Adams (4), Frank
- Person: Adams, John Couch
- Person: Adamson, Iain Thomas Arthur Carpenter
- Person: Adelard Of Bath
- Person: Adian, Sergei Ivanovich
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- Person: Valiant, Leslie Gabriel
- Person: Valiron, Georges Jean Marie
- Person: Van Amringe, John Howard
- Person: Van Ceulen, Ludolph
- Person: Van Dantzig, David
- Person: Van Heuraet, Hendrik
- Person: Van Kampen, Egbert Rudolf
- Person: Van Lansberge, Johan Philip
- Person: Van Lint, Jacobus Hendricus
- Person: Van Roomen, Adriaan
- Person: Van Schooten, Frans
- Person: Van Vleck, Edward Burr
- Person: Vandermonde, Alexandre-Théophile
- Person: Vandiver, Harry Schultz
- Person: Vanstone, Ray
- Person: Varadhan, Sathamangalam Ranga Iyengar Srinivasa
- Person: Varahamihira
- Person: Varga (2), Richard Steven
- Person: Varga, Ottó
- Person: Varignon, Pierre
- Person: Varopoulos, Theodoros
- Person: Vashchenko-Zakharchenko, Mikhail Egorovich
- Person: Veblen, Oswald
- Person: Vekua, Ilya Nestorovich
- Person: Velez-Rodriguez, Argelia
- Person: Venn, John
- Person: Verbiest, Ferdinand
- Person: Verblunsky, Samuel
- Person: Verhulst, Pierre François
- Person: Vernier, Pierre
- Person: Vernon, Siobhan O'Shea
- Person: Veronese, Giuseppe
- Person: Verrier, Urbain Jean Joseph Le
- Person: Vessiot, Ernest-Paulin-Joseph
- Person: Viazovska, Maryna
- Person: Vidav, Ivan
- Person: Vietoris, Leopold
- Person: Vijayanandi
- Person: Vilant, Nicolas
- Person: Vince, Samuel
- Person: Vinogradov, Ivan Matveevich
- Person: Vinti, Calogero
- Person: Vitali, Giuseppe
- Person: Vitushkin, Anatoli Georgievich
- Person: Vivanti, Giulio
- Person: Viviani, Vincenzo
- Person: Viète, François
- Person: Vlacq, Adriaan
- Person: Vladimirov, Vasilii Sergeevich
- Person: Voevodsky, Vladimir Aleksandrovich
- Person: Volterra, Samuel Giuseppe Vito
- Person: Von Brill, Alexander Wilhelm
- Person: Von Dyck, Walther Franz Anton
- Person: Von Escherich, Gustav
- Person: Von Helmholtz, Hermann Ludwig Ferdinand
- Person: Von Koch, Niels Fabian Helge
- Person: Von Kármán, Theodore
- Person: Von Leibniz, Gottfried Wilhelm
- Person: Von Lindemann, Carl Louis Ferdinand
- Person: Von Mises, Richard
- Person: Von Nagel, Christian Heinrich
- Person: Von Neumann, John
- Person: Von Segner, Johann Andreas
- Person: Von Seidel, Philipp Ludwig
- Person: Von Staudt, Karl Georg Christian
- Person: Von Tschirnhaus, Ehrenfried Walter
- Person: Von Vega, Georg Freiherr
- Person: Voronoy, Georgy Fedoseevich
- Person: Vrănceanu, Gheorghe
- Person: Vályi, Gyula
- Person: Waerden, Bartel Leendert van der
- Person: Wald, Abraham
- Person: Wales, Muriel Kennett
- Person: Walfisz, Arnold
- Person: Walker (2), Gilbert
- Person: Walker (3), Geoffrey
- Person: Walker, John James
- Person: Wall, Hubert
- Person: Wallace (2), Alexander Doniphan
- Person: Wallace, William
- Person: Waller, Derek Arthur
- Person: Wallis, John
- Person: Walsh (2), Joseph
- Person: Walsh, John
- Person: Walter, Marion Ilse
- Person: Walton, Ernest
- Person: Wang, Hsien Chung
- Person: Wangerin, Friedrich Heinrich Albert
- Person: Wantzel, Pierre Laurent
- Person: Ward, Seth
- Person: Warga, Jack
- Person: Waring, Edward
- Person: Warner, Mary Wynne
- Person: Warschawski, Stefan E
- Person: Washington, Talitha
- Person: Waterston, John James
- Person: Watson (2), William
- Person: Watson (3), Henry William
- Person: Watson (4), George Neville
- Person: Watson, Henry
- Person: Watt, James
- Person: Wattie, James MacPherson
- Person: Wavre, Rolin
- Person: Wazewski, Tadeusz
- Person: Weatherburn, Charles Ernest
- Person: Weatherhead, Kenneth Kilpatrick
- Person: Weaver, Warren
- Person: Weber (2), Heinrich
- Person: Weber (3), Heinrich Friedrich
- Person: Weber (4), Johanna
- Person: Weber, Wilhelm Eduard
- Person: Wedderburn, Joseph Henry Maclagen
- Person: Wegner, Udo Hugo Helmuth
- Person: Weierstrass, Karl Theodor Wilhelm
- Person: Weil, André Abraham
- Person: Weiler, Adolf
- Person: Weingarten, Julius
- Person: Weinstein, Alexander
- Person: Weisbach, Julius Lugwig
- Person: Weise, Karl Heinrich
- Person: Welchman, William Gordon
- Person: Weldon, Walter Frank Raphael
- Person: Wending, Mei
- Person: Wenninger, Magnus Joseph
- Person: Werner (2), Wendelin
- Person: Werner, Johann
- Person: Wessel, Caspar
- Person: West, John
- Person: Weyl, Hermann Klaus Hugo
- Person: Weyr (2), Eduard
- Person: Weyr, Emil
- Person: Wheeler, Anna Johnson Pell
- Person: Whewell, William
- Person: Whish, Charles Matthew
- Person: Whiston, William
- Person: White, Henry Seely
- Person: Whitehead (2), Henry
- Person: Whitehead, Alfred North
- Person: Whiteside, Derek Thomas
- Person: Whitney, Hassler
- Person: Whitrow, Gerald
- Person: Whittaker (2), John
- Person: Whittaker, Edmund Taylor
- Person: Whitworth, Allen
- Person: Whyburn (2), William Marvin
- Person: Whyburn, William
- Person: Widder, David
- Person: Widman, Johannes
- Person: Wiegold, James
- Person: Wielandt, Helmut
- Person: Wiener (2), Hermann
- Person: Wiener (3), Ludwig Christian
- Person: Wiener (4), Norbert
- Person: Wiener, Christian
- Person: Wigner, Eugene Paul
- Person: Wilczynski, Ernest Julius
- Person: Wilder, Raymond Louis
- Person: Wiles, Andrew John
- Person: Wilf, Herbert Saul
- Person: Wilkins (2), J. Ernest
- Person: Wilkins, John
- Person: Wilkinson, James Hardy
- Person: Wilks, Samuel Stanley
- Person: William Of Ockham
- Person: Williams (2), Evan James
- Person: Williams (3), Lloyd
- Person: Williams, William Lloyd Garrison
- Person: Williamson, John
- Person: Wilson (2), John
- Person: Wilson (3), John
- Person: Wilson (4), Edwin
- Person: Wilson (5), Bertram
- Person: Wilson, Alexander
- Person: Wiltheiss, Ernst Eduard
- Person: Wilton, John Raymond
- Person: Wiman, Anders
- Person: Winkler, Wilhelm
- Person: Wintner, Aurel Friedrich
- Person: Wirtinger, Wilhelm
- Person: Wishart, John
- Person: Witt, Ernst
- Person: Witten, Edward
- Person: Wittgenstein, Ludwig Josef Johann
- Person: Wittich, Paul
- Person: Wolf (2), František
- Person: Wolf, Johann Rudolf
- Person: Wolfowitz, Jacob
- Person: Wolstenholme, Joseph
- Person: Womersley, John
- Person: Wood, Frances Chick
- Person: Woodard, Dudley Weldon
- Person: Woodhouse, Robert
- Person: Woods, Leslie Colin
- Person: Woodward, Robert Simpson
- Person: Wren (2), Thomas Lancaster
- Person: Wren, Sir Christopher
- Person: Wright (2), Edward Maitland
- Person: Wright (3), Fred
- Person: Wright, Sewall Green
- Person: Wrinch, Dorothy Maud
- Person: Wschebor, Mario
- Person: Wu (2), Sijue
- Person: Wu, Wen-Tsun
- Person: Wussing, Hans
- Person: Wylie, Shaun
- Person: Xenocrates Of Chalcedon
- Person: Xian, Jia
- Person: Xiaotong, Wang
- Person: Yamabe, Hidehiko
- Person: Yang, Xiahou
- Person: Yano, Kentaro
- Person: Yasuaki, Aida
- Person: Yates, Frank
- Person: Yativrsabha
- Person: Yau, Shing-Tung
- Person: Yavanesvara
- Person: Yoccoz, Jean-Christophe
- Person: Yosida, Kosaku
- Person: Youden, William John
- Person: Young (2), William Henry
- Person: Young (3), Grace Chisholm
- Person: Young (4), Alfred
- Person: Young (5), Andrew
- Person: Young (6), Laurence Chisholm
- Person: Young (7), Lai-Sang
- Person: Young, Thomas
- Person: Youqin, Zhao
- Person: Yuan (2), Wang
- Person: Yuan, Ruan
- Person: Yue, Xu
- Person: Yule, George Udny
- Person: Yushkevich, Adolph Andrei Pavlovich
- Person: Zaanen, Adriaan Cornelis
- Person: Zalts, Karlis
- Person: Zarankiewicz, Kazimierz
- Person: Zaremba, Stanislaw
- Person: Zariski, Oscar
- Person: Zassenhaus, Hans Julius
- Person: Zeckendorf, Édouard
- Person: Zeeman, Erik Christopher
- Person: Zehfuss, Georg
- Person: Zelmanov, Efim Isaakovich
- Person: Zeno Of Elea
- Person: Zeno Of Sidon
- Person: Zenodorus
- Person: Zermelo, Ernst Friedrich Ferdinand
- Person: Zervos, Panagiotis
- Person: Zeuthen, Hieronymous Georg
- Person: Zhautykov, Orymbek
- Person: Zhi, Li
- Person: Zhukovsky, Nikolai Egorovich
- Person: Zi, Sun
- Person: Zippin, Leo
- Person: Zolotarev, Egor Ivanovich
- Person: Zorn, Max August
- Person: Zuse, Konrad
- Person: Zwicky, Fritz
- Person: Zygalski, Henryk
- Person: Zygmund, Antoni Szczepan
- Person: Ásgeirsson, Leifur
- Person: Ávila, Artur
- Person: Öpik, Ernst
- Person: Čech, Eduard
- Person: Łoś, Jerzy Maria Michał
- Person: Łukasiewicz, Jan
- Person: Ślebarski, Tadeusz Boleslaw
- Person: Świerczkowski, Stanisław
- Person: Żorawski, Kazimierz
- Person: Żyliński, Eustachy
- Person: Țarină, Marian
- Problem: "Strand" Patience
- Problem: A Bank Holiday Puzzle
- Problem: A Calendar Puzzle
- Problem: A Census Puzzle
- Problem: A Chain Puzzle
- Problem: A Charitable Bequest
- Problem: A Chessboard Fallacy
- Problem: A Cutting-out Puzzle
- Problem: A Deal in Apples
- Problem: A Deal in Eggs
- Problem: A Dormitory Puzzle
- Problem: A Dungeon Puzzle
- Problem: A Family Party
- Problem: A Fence Problem
- Problem: A Juvenile Puzzle
- Problem: A Kite-flying Puzzle
- Problem: A Legal Difficulty
- Problem: A Lodging-house Difficulty
- Problem: A Magic Square Of Composites
- Problem: A Match Mystery
- Problem: A Mixed Pedigree
- Problem: A New Bishop's Puzzle
- Problem: A New Counter Puzzle
- Problem: A New Match Puzzle
- Problem: A New Money Puzzle
- Problem: A Packing Puzzle
- Problem: A Plantation Puzzle
- Problem: A Postoffice Perplexity
- Problem: A Printer's Error
- Problem: A Problem in Mosaics
- Problem: A Problem in Squares
- Problem: A Puzzle For Motorists
- Problem: A Puzzle With Pawns
- Problem: A Puzzle for Card-players
- Problem: A Puzzle in Reversals
- Problem: A Puzzling Legacy
- Problem: A Puzzling Watch
- Problem: A Queer Coincidence
- Problem: A Queer Thing in Money
- Problem: A Question of Definition
- Problem: A Railway Muddle
- Problem: A Railway Puzzle
- Problem: A Shopping Perplexity
- Problem: A Study in Thrift
- Problem: A Tangram Paradox
- Problem: A Tennis Tournament
- Problem: A Time Puzzle
- Problem: A Trick With Dice
- Problem: A War Puzzle Game
- Problem: A Wonderful Village
- Problem: Academic Courtesies
- Problem: Adding The Digits
- Problem: An Acrostic Puzzle
- Problem: An Amazing Dilemma
- Problem: An Easy Dissection Puzzle
- Problem: An Easy Square Puzzle
- Problem: An Episcopal Visitation
- Problem: Ancient Chinese Puzzle
- Problem: Another Joiner's Problem
- Problem: Another Linoleum Puzzle
- Problem: Another Patchwork Puzzle
- Problem: Applications of the Jacobi Symbol
- Problem: Are there infinitely many primorial primes?
- Problem: Arranging The Jampots
- Problem: At a Cattle Market
- Problem: Average Speed
- Problem: Awkward Money
- Problem: Bachet's Square
- Problem: Beef and Sausages
- Problem: Bishops in Convocation
- Problem: Bishops — guarded
- Problem: Bishops — unguarded
- Problem: Boards With An Odd Number Of Squares
- Problem: Boys And Girls
- Problem: Broken Items in the Box
- Problem: Broken Items in the Box II
- Problem: Building The Tetrahedron
- Problem: Buying Apples
- Problem: Buying Chestnuts
- Problem: Buying Presents
- Problem: Calculating Quadratic Residues
- Problem: Calculating an Infinite Sum
- Problem: Card Magic Squares
- Problem: Card Triangles
- Problem: Catching The Mice
- Problem: Catching the Thief
- Problem: Central Solitaire
- Problem: Changing Places
- Problem: Checking if `$K_5$` is Planar
- Problem: Checking if `$K_{3,3}$` is Planar
- Problem: Checkmate!
- Problem: Chequered Board Divisions
- Problem: Cherries And Plums
- Problem: Chessboard Solitaire
- Problem: Circling the Squares
- Problem: Concerning Tommy's Age
- Problem: Concerning Wheels
- Problem: Counter Crosses
- Problem: Counter Solitaire
- Problem: Counting The Rectangles
- Problem: Crossing The River Axe
- Problem: Crossing The Stream
- Problem: Curious Numbers
- Problem: Defective Observation
- Problem: Digital Division
- Problem: Digital Multiplication
- Problem: Digital Square Numbers
- Problem: Digits and Squares
- Problem: Dissecting a Mittre
- Problem: Domestic Economy
- Problem: Dominoes In Progression
- Problem: Donkey Riding
- Problem: Drawing A Spiral
- Problem: Drawing Her Pension
- Problem: Eight Jolly Gaol Birds
- Problem: Even Perfect Numbers Problem
- Problem: Exercise For Prisoners
- Problem: Farmer Lawrence's Cornfields
- Problem: Farmer Wurzel's Estate
- Problem: Fifteen Letter Puzzle
- Problem: Find Ada's Surname
- Problem: Find The Man's Wife
- Problem: Five Jealous Husbands
- Problem: Giving Change
- Problem: Gold Packing In Russia
- Problem: Hannah's Puzzle
- Problem: He Banner Puzzle
- Problem: Heads or Tails
- Problem: Heard on the Tube Railway
- Problem: How Old Was Mary?
- Problem: How To Draw An Oval
- Problem: How To Make Cisterns
- Problem: Immovable Pawns
- Problem: Indiscriminate Charity
- Problem: Inspecting A Mine
- Problem: Interpolating Numbers With a Polynomial
- Problem: Jack And The Beanstalk
- Problem: Judkins's Cattle
- Problem: King Arthur's Knights
- Problem: Lady Belinda's Garden
- Problem: Linoleum Cutting
- Problem: Lion-hunting
- Problem: Lions And Crowns
- Problem: Magic Squares Of Two Degrees
- Problem: Mamma's Age
- Problem: Mary and Marmaduke
- Problem: Mixing The Tea
- Problem: More Mixed Fractions
- Problem: Mother and Daughter
- Problem: Mr. Gubbins in a Fog
- Problem: Mrs. Hobson's Hearthrug
- Problem: Mrs. Perkins's Quilt
- Problem: Mrs. Smiley's Christmas Present
- Problem: Mrs. Timpkin's Age
- Problem: New Measuring Puzzle
- Problem: Next-door Neighbors
- Problem: Nine Jolly Gaol Birds
- Problem: Odd And Even Digits
- Problem: Odd Perfect Numbers Problem
- Problem: Painting A Pyramid
- Problem: Painting The Die
- Problem: Painting the Lamp-Posts
- Problem: Papa's Puzzle
- Problem: Pheasant-shooting
- Problem: Placing Halfpennies
- Problem: Plates And Coins
- Problem: Pocket Money
- Problem: Puss In The Corner
- Problem: Queens And Bishop Puzzle
- Problem: Queer Chess
- Problem: Queer Multiplication
- Problem: Queer Relationships
- Problem: Rackbrane's Little Loss
- Problem: Reeping the Corn
- Problem: Round The Coast
- Problem: Rover's Age
- Problem: Saturday Marketing
- Problem: Setting The Board
- Problem: Simple Division
- Problem: Simple Multiplication
- Problem: Sir Edwyn De Tudor
- Problem: Slow Cricket
- Problem: Square Money
- Problem: St. George And The Dragon
- Problem: St. George's Banner
- Problem: Stalemate
- Problem: Stealing The Bell-ropes
- Problem: Stealing The Castle Treasure
- Problem: Such A Getting Upstairs
- Problem: Sum of Consecutive Positive Integers
- Problem: Sum of Consecutive Squares
- Problem: Sums of Falling Factorial Powers
- Problem: Sums of Powers of Two
- Problem: TSP - The Traveling Salesman Problem
- Problem: The "T" Card Puzzle
- Problem: The Abbot's Puzzle
- Problem: The Abbot's Window
- Problem: The Amazons
- Problem: The Antiquary's Chain
- Problem: The Artilleryman's Dilemma
- Problem: The Bag of Nuts
- Problem: The Ball Problem
- Problem: The Banker's Puzzle
- Problem: The Barrel Puzzle
- Problem: The Barrel of Beer
- Problem: The Barrels Of Balsam
- Problem: The Barrels Of Honey
- Problem: The Basket Of Potatoes
- Problem: The Baskets Of Plums
- Problem: The Battle of Hastings
- Problem: The Beanfeast Puzzle
- Problem: The Betsy Ross Puzzle
- Problem: The Bicylce Thief
- Problem: The Board In Compartments
- Problem: The Broken Coins
- Problem: The Bun Puzzle
- Problem: The Burmese Plantation
- Problem: The Cab Numbers
- Problem: The Card Frame Puzzle
- Problem: The Cardboard Box
- Problem: The Cardboard Chain
- Problem: The Century Puzzle
- Problem: The Chessboard Sentence
- Problem: The Chinese Chessboard
- Problem: The Chocolate Squares
- Problem: The Christmas Pudding
- Problem: The Christmas-Boxes
- Problem: The Cigar Puzzle
- Problem: The City Luncheons
- Problem: The Clothes Line Puzzle
- Problem: The Club Clock
- Problem: The Coloured Counters
- Problem: The Compasses Puzzle
- Problem: The Cone Puzzle
- Problem: The Converted Miser
- Problem: The Costermonger's Puzzle
- Problem: The Crescent Puzzle
- Problem: The Cross Of Cards
- Problem: The Cross Target
- Problem: The Cross and the Triangle
- Problem: The Crowded Chessboard
- Problem: The Crusader
- Problem: The Cubic Knight's Tour
- Problem: The Cushion Covers
- Problem: The Cyclists' Feast
- Problem: The Cyclists' Tour
- Problem: The Deified Puzzle
- Problem: The Diamond Puzzle
- Problem: The Dice Numbers
- Problem: The Digital Century
- Problem: The Dissected Circle
- Problem: The Dissected Triangle
- Problem: The Doctor's Query
- Problem: The Domino Frame Puzzle
- Problem: The Dovetailed Block
- Problem: The Dutchmen's Wives
- Problem: The Eccentric Cheesemonger
- Problem: The Educated Frogs
- Problem: The Eight Engines
- Problem: The Eight Queens
- Problem: The Eight Rooks
- Problem: The Eight Stars
- Problem: The Eight Sticks
- Problem: The Eight Villas
- Problem: The Eighteen Dominoes
- Problem: The Exchange Puzzle
- Problem: The Excursion Ticket Puzzle
- Problem: The Family Ages
- Problem: The Farmer and His Sheep
- Problem: The Fifteen Dominoes
- Problem: The Fifteen Turnings
- Problem: The Five Brigands
- Problem: The Five Crescents of Byzantium
- Problem: The Five Dogs Puzzle
- Problem: The Five Dominoes
- Problem: The Five Pennies
- Problem: The Fly On The Octahedron
- Problem: The Folded Cross
- Problem: The Football Players
- Problem: The Forsaken King
- Problem: The Forty-nine Counters
- Problem: The Forty-nine Stars
- Problem: The Four Elopements
- Problem: The Four Frogs
- Problem: The Four Kangaroos
- Problem: The Four Knights' Tours
- Problem: The Four Lions
- Problem: The Four Postage Stamps
- Problem: The Four Sevens
- Problem: The Four Sons
- Problem: The Garden Puzzle
- Problem: The Garden Walls
- Problem: The Gardener And The Cook
- Problem: The Gentle Art Of Stamp-licking
- Problem: The Glass Balls
- Problem: The Grand Lama's Problem
- Problem: The Grand Tour
- Problem: The Grasshopper Puzzle
- Problem: The Great Monad
- Problem: The Great Scramble
- Problem: The Greyhound Puzzle
- Problem: The Grocer and Draper
- Problem: The Hat Puzzle
- Problem: The Hat-peg Puzzle
- Problem: The Honest Dairyman
- Problem: The Honeycomb Puzzle
- Problem: The Horse-race Puzzle
- Problem: The Hydroplane Question
- Problem: The Hymn-board Poser
- Problem: The Icosahedron Puzzle
- Problem: The Industrious Bookworm
- Problem: The Joiner's Problem
- Problem: The Junior Clerk's Puzzle
- Problem: The Keg Of Wine
- Problem: The Kennel Puzzle
- Problem: The King And The Castles
- Problem: The Knight-guards
- Problem: The Laborer's Puzzle
- Problem: The Landowner's Fences
- Problem: The Languishing Maiden
- Problem: The Leap-Year Ladies
- Problem: The Letter Block Puzzle
- Problem: The Level Puzzle
- Problem: The Lion And The Man
- Problem: The Lockers Puzzle
- Problem: The Magic Knight's Tour
- Problem: The Magic Strips
- Problem: The Mandarin's "T" Puzzle
- Problem: The Mandarin's Puzzle
- Problem: The Market Women
- Problem: The Milkmaid Puzzle
- Problem: The Millionaire's Perplexity
- Problem: The Miners' Holiday
- Problem: The Monk And The Bridges
- Problem: The Monstrosity
- Problem: The Montenegrin Dice Game
- Problem: The Motor-car Race
- Problem: The Motor-car Tour
- Problem: The Motor-garage Puzzle
- Problem: The Mouse-trap Puzzle
- Problem: The Muddletown Election
- Problem: The Mystic Eleven
- Problem: The Nine Almonds
- Problem: The Nine Counters
- Problem: The Nine Schoolboys
- Problem: The Nine Treasure Boxes
- Problem: The Number Checks Puzzle
- Problem: The Parish Council Election
- Problem: The Passenger's Fare
- Problem: The Peal Of Bells
- Problem: The Pebble Game
- Problem: The Pentagon and Square
- Problem: The Pierrot's Puzzle
- Problem: The Potato Puzzle
- Problem: The Puzzle Wall
- Problem: The Puzzling Money-Boxes
- Problem: The Queen's Journey
- Problem: The Queen's Tour
- Problem: The Railway Station Clock
- Problem: The Rook's Journey
- Problem: The Rook's Tour
- Problem: The Rookery
- Problem: The Round Table
- Problem: The Ruby Brooch
- Problem: The Sabbath Puzzle
- Problem: The Sailor's Puzzle
- Problem: The Scientific Skater
- Problem: The Sculptor's Problem
- Problem: The See-saw Puzzle
- Problem: The Seven Pigs
- Problem: The Sheep-fold
- Problem: The Siberian Dungeons
- Problem: The Silk Patchwork
- Problem: The Six Frogs
- Problem: The Six Pawns
- Problem: The Six Sheep-pens
- Problem: The Sixteen Sheep
- Problem: The Southern Cross
- Problem: The Spanish Dungeon
- Problem: The Spanish Miser
- Problem: The Spot on the Table
- Problem: The Square of Veneer
- Problem: The Squares Of Brocade
- Problem: The Star Puzzle
- Problem: The Stonemason's Problem
- Problem: The Stop-Watch
- Problem: The Suffragists' Meeting
- Problem: The Sultan's Army
- Problem: The Table-top and Stools
- Problem: The Ten Apples
- Problem: The Ten Coins
- Problem: The Ten Counters
- Problem: The Ten Prisoners
- Problem: The Tethered Goat
- Problem: The Thirty-Three Pearls
- Problem: The Thirty-six Letter Blocks
- Problem: The Three Clocks
- Problem: The Three Groups
- Problem: The Three Railway Stations
- Problem: The Three Sheep
- Problem: The Three Villages
- Problem: The Tiring Irons
- Problem: The Torn Number
- Problem: The Troublesome Eight
- Problem: The Trusses of Hay
- Problem: The Tube Inspector's Puzzle
- Problem: The Twelve Mince-pies
- Problem: The Twelve Pennies
- Problem: The Twenty-one Trees
- Problem: The Twickenham Puzzle
- Problem: The Two Aeroplanes
- Problem: The Two Horseshoes
- Problem: The Two Pawns
- Problem: The Two Rooks
- Problem: The Two Trains
- Problem: The Union Jack
- Problem: The Victoria Cross Puzzle
- Problem: The Village Cricket Match
- Problem: The Village Simpleton
- Problem: The Voters' Puzzle
- Problem: The Wapshaw's Wharf Mystery
- Problem: The Wassail Bowl
- Problem: The Widow's Legacy
- Problem: The Wizard's Cats
- Problem: The Wrong Hats
- Problem: The Yacht Race
- Problem: The Yorkshire Estates
- Problem: The new Year's Eve Suppers
- Problem: Their Ages
- Problem: Thirty-six Mates
- Problem: Those Fifteen Sheep
- Problem: Three Men In A Boat
- Problem: To Construct a Partition of a Given Set
- Problem: Torpedo Practice
- Problem: Turks And Russians
- Problem: Twin Prime Problem
- Problem: Two Crosses From One
- Problem: Two New Magic Squares
- Problem: Two Questions in Probabilities
- Problem: Under the Veil
- Problem: Verifying Group Properties
- Problem: Verifying Subgroup Properties
- Problem: Visiting The Towns
- Problem: Water, Gas, And Electricity
- Problem: What Was the Time?
- Problem: Who Was First?
- Problem: Wilson's Poser
- Problem: Wine And Water
- Problem: Youthful Precocity
- Proof: (related to Corollary: Abelian Group of Vectors Under Addition)
- Proof: (related to Corollary: Barycentric Coordinates, Barycenter)
- Proof: (related to Corollary: General Associative Law)
- Proof: (related to Corollary: General Commutative Law)
- Proof: (related to Corollary: Intersection of Convex Affine Sets)
- Proof: (related to Corollary: Properties of a Real Scalar Product)
- Proof: (related to Corollary: Rules for Exponentiation in a Group)
- Proof: (related to Corollary: Solutions of a Linear Equation with many Unknowns)
- Proof: (related to Lemma: Any Positive Characteristic Is a Prime Number)
- Proof: (related to Lemma: Cyclic Groups are Abelian)
- Proof: (related to Lemma: Divisibility of Principal Ideals)
- Proof: (related to Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE)
- Proof: (related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
- Proof: (related to Lemma: Factor Groups)
- Proof: (related to Lemma: Factor Rings, Generalization of Congruence Classes)
- Proof: (related to Lemma: Fiber of Maximal Ideals)
- Proof: (related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
- Proof: (related to Lemma: Fiber of Prime Ideals)
- Proof: (related to Lemma: Greatest Common Divisor and Least Common Multiple of Ideals)
- Proof: (related to Lemma: Group Homomorphisms and Normal Subgroups)
- Proof: (related to Lemma: Kernel and Image of Group Homomorphism)
- Proof: (related to Lemma: Kernel and Image of a Group Homomorphism are Subgroups)
- Proof: (related to Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring)
- Proof: (related to Lemma: Prime Ideals of Multiplicative Systems in Integral Domains)
- Proof: (related to Lemma: Subgroups and Their Cosets are Equipotent)
- Proof: (related to Lemma: Subgroups of Cyclic Groups)
- Proof: (related to Lemma: Uniqueness Lemma of a Finite Basis)
- Proof: (related to Proposition: A Field with an Absolute Value is a Metric Space)
- Proof: (related to Proposition: Abelian Group of Matrices Under Addition)
- Proof: (related to Proposition: Additive Subgroups of Integers)
- Proof: (related to Proposition: Cancellation Law)
- Proof: (related to Proposition: Characterization of Dependent Absolute Values)
- Proof: (related to Proposition: Characterization of Non-Archimedean Absolute Values)
- Proof: (related to Proposition: Criteria for Subgroups)
- Proof: (related to Proposition: Finite Order of an Element Equals Order Of Generated Group)
- Proof: (related to Proposition: Group Homomorphisms with Cyclic Groups)
- Proof: (related to Proposition: Group of Units)
- Proof: (related to Proposition: In a Field, `$0$` Is Unequal `$1$`)
- Proof: (related to Proposition: Open and Closed Subsets of a Zariski Topology)
- Proof: (related to Proposition: Properties of Cosets)
- Proof: (related to Proposition: Properties of a Complex Scalar Product)
- Proof: (related to Proposition: Properties of a Group Homomorphism)
- Proof: (related to Proposition: Quadratic Formula)
- Proof: (related to Proposition: Quotient Space)
- Proof: (related to Proposition: Simple Calculations Rules in a Group)
- Proof: (related to Proposition: Simple Consequences from the Definition of a Vector Space)
- Proof: (related to Proposition: Spectrum Function of Commutative Rings)
- Proof: (related to Proposition: Square of a Non-Zero Element is Positive in Ordered Fields)
- Proof: (related to Proposition: Subgroups of Finite Cyclic Groups)
- Proof: (related to Proposition: Subset of Powers is a Submonoid)
- Proof: (related to Proposition: Unique Solvability of `$a\ast x=b$` in Groups)
- Proof: (related to Proposition: `$0$` Is Less Than `$1$` In Ordered Fields)
- Proof: (related to Theorem: Classification of Cyclic Groups)
- Proof: (related to Theorem: Classification of Finite Groups with the Order of a Prime Number)
- Proof: (related to Theorem: Connection between Rings, Ideals, and Fields)
- Proof: (related to Theorem: Construction of Fields from Integral Domains)
- Proof: (related to Theorem: Construction of Groups from Commutative and Cancellative Semigroups)
- Proof: (related to Theorem: Finite Basis Theorem)
- Proof: (related to Theorem: Finite Integral Domains are Fields)
- Proof: (related to Theorem: First Isomorphism Theorem for Groups)
- Proof: (related to Theorem: Order of Cyclic Group (Fermat's Little Theorem))
- Proof: (related to Theorem: Order of Subgroup Divides Order of Finite Group)
- Proof: (related to Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs)
- Proof: (related to Corollary: (Real) Exponential Function Is Always Positive)
- Proof: (related to Corollary: All Uniformly Continuous Functions are Continuous)
- Proof: (related to Corollary: All Zeros of Cosine and Sine)
- Proof: (related to Corollary: Arguments for which Cosine and Sine are Equal to Each Other)
- Proof: (related to Corollary: Closed Real Intervals Are Compact)
- Proof: (related to Corollary: Continuous Functions Mapping Compact Domains to Real Numbers are Bounded)
- Proof: (related to Corollary: Continuous Real Functions on Closed Intervals are Bounded)
- Proof: (related to Corollary: Convergence of Complex Conjugate Sequence)
- Proof: (related to Corollary: Cosine and Sine are Periodic Functions)
- Proof: (related to Corollary: Derivative of a Constant Function)
- Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
- Proof: (related to Corollary: Derivative of a Linear Function `\(ax+b\)`)
- Proof: (related to Corollary: Differentiable Functions are Continuous)
- Proof: (related to Corollary: Estimating the Growth of a Function with its Derivative)
- Proof: (related to Corollary: Exchanging the Limit of Function Values with the Function Value of the Limit of Arguments)
- Proof: (related to Corollary: Exponential Function Is Non-Negative (Real Case))
- Proof: (related to Corollary: Exponential Function Is Strictly Monotonically Increasing)
- Proof: (related to Corollary: Exponential Function and the Euler Constant)
- Proof: (related to Corollary: Functions Continuous at a Point and Identical to Other Functions in a Neighborhood of This Point)
- Proof: (related to Corollary: Limit of N-th Roots)
- Proof: (related to Corollary: Monotony Criterion for Absolute Series)
- Proof: (related to Corollary: More Insight to Euler's Identity)
- Proof: (related to Corollary: Negative Cosine and Sine vs Shifting the Argument)
- Proof: (related to Corollary: Non-Cauchy Sequences are Not Convergent)
- Proof: (related to Corollary: Real Numbers Can Be Approximated by Rational Numbers)
- Proof: (related to Corollary: Real Polynomials of Odd Degree Have at Least One Real Root)
- Proof: (related to Corollary: Reciprocity of Complex Exponential Function, Non-Zero Property)
- Proof: (related to Corollary: Reciprocity of Exponential Function of General Base, Non-Zero Property)
- Proof: (related to Corollary: Reciprocity of Exponential Function, Non-Zero Property)
- Proof: (related to Corollary: Representing Real Cosine by Complex Exponential Function)
- Proof: (related to Corollary: Representing Real Sine by Complex Exponential Function)
- Proof: (related to Corollary: Sufficient Condition for a Function to be Constant)
- Proof: (related to Corollary: Taylor's Formula for Polynomials)
- Proof: (related to Corollary: Value of Zero to the Power of X)
- Proof: (related to Lemma: Abel's Lemma for Testing Convergence)
- Proof: (related to Lemma: Addition and Scalar Multiplication of Riemann Upper and Lower Integrals)
- Proof: (related to Lemma: Approximability of Continuous Real Functions On Closed Intervals By Step Functions)
- Proof: (related to Lemma: Convergence Test for Telescoping Series)
- Proof: (related to Lemma: Decreasing Sequence of Suprema of Extended Real Numbers)
- Proof: (related to Lemma: Euler's Identity)
- Proof: (related to Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point)
- Proof: (related to Lemma: Increasing Sequence of Infima of Extended Real Numbers)
- Proof: (related to Lemma: Invertible Functions on Real Intervals)
- Proof: (related to Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine)
- Proof: (related to Lemma: Sum of Roots Of Unity in Complete Residue Systems)
- Proof: (related to Lemma: Trapezoid Rule)
- Proof: (related to Lemma: Unit Circle)
- Proof: (related to Lemma: Upper Bound for the Product of General Powers)
- Proof: (related to Proposition: A General Criterion for the Convergence of Infinite Complex Series)
- Proof: (related to Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions)
- Proof: (related to Proposition: Abel's Test)
- Proof: (related to Proposition: Additivity Theorem of Tangent)
- Proof: (related to Proposition: Additivity Theorems of Cosine and Sine)
- Proof: (related to Proposition: Approximation of Functions by Taylor's Formula)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Difference)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Division)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Product)
- Proof: (related to Proposition: Arithmetic of Functions with Limits - Sums)
- Proof: (related to Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule)
- Proof: (related to Proposition: Bounds for Partial Sums of Exponential Series)
- Proof: (related to Proposition: Calculation Rules for General Powers)
- Proof: (related to Proposition: Calculations with Uniformly Convergent Functions)
- Proof: (related to Proposition: Cauchy Condensation Criterion)
- Proof: (related to Proposition: Cauchy Criterion)
- Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Complex Series)
- Proof: (related to Proposition: Cauchy Product of Absolutely Convergent Series)
- Proof: (related to Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent)
- Proof: (related to Proposition: Cauchy-Schwarz Inequality for Integral p-norms)
- Proof: (related to Proposition: Cauchy-Schwarz Test)
- Proof: (related to Proposition: Cauchy–Schwarz Inequality)
- Proof: (related to Proposition: Chain Rule)
- Proof: (related to Proposition: Characterization of Monotonic Functions via Derivatives)
- Proof: (related to Proposition: Closed Formula for the Maximum and Minimum of Two Numbers)
- Proof: (related to Proposition: Closed Subsets of Compact Sets are Compact)
- Proof: (related to Proposition: Closed n-Dimensional Cuboids Are Compact)
- Proof: (related to Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum)
- Proof: (related to Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed)
- Proof: (related to Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth)
- Proof: (related to Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences)
- Proof: (related to Proposition: Complex Conjugate of Complex Exponential Function)
- Proof: (related to Proposition: Complex Convergent Sequences are Bounded)
- Proof: (related to Proposition: Complex Exponential Function)
- Proof: (related to Proposition: Composition of Continuous Functions at a Single Point)
- Proof: (related to Proposition: Compositions of Continuous Functions on a Whole Domain)
- Proof: (related to Proposition: Continuity of Complex Exponential Function)
- Proof: (related to Proposition: Continuity of Cosine and Sine)
- Proof: (related to Proposition: Continuity of Exponential Function of General Base)
- Proof: (related to Proposition: Continuity of Exponential Function)
- Proof: (related to Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals)
- Proof: (related to Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable)
- Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity)
- Proof: (related to Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero)
- Proof: (related to Proposition: Convergence Behavior of the Sequence `\((b^n)\)`)
- Proof: (related to Proposition: Convergence Behaviour of Absolutely Convergent Series)
- Proof: (related to Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero)
- Proof: (related to Proposition: Convergent Complex Sequences Are Bounded)
- Proof: (related to Proposition: Convergent Complex Sequences Are Cauchy Sequences)
- Proof: (related to Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences)
- Proof: (related to Proposition: Convergent Real Sequences Are Cauchy Sequences)
- Proof: (related to Proposition: Convergent Real Sequences are Bounded)
- Proof: (related to Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space)
- Proof: (related to Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space)
- Proof: (related to Proposition: Convergent Sequences are Bounded)
- Proof: (related to Proposition: Convex Functions on Open Intervals are Continuous)
- Proof: (related to Proposition: Convexity and Concaveness Test)
- Proof: (related to Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm)
- Proof: (related to Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`)
- Proof: (related to Proposition: Derivative of Cosine)
- Proof: (related to Proposition: Derivative of General Powers of Positive Numbers)
- Proof: (related to Proposition: Derivative of Sine)
- Proof: (related to Proposition: Derivative of Tangent)
- Proof: (related to Proposition: Derivative of an Invertible Function on Real Invervals)
- Proof: (related to Proposition: Derivative of the Exponential Function)
- Proof: (related to Proposition: Derivative of the Inverse Sine)
- Proof: (related to Proposition: Derivative of the Inverse Tangent)
- Proof: (related to Proposition: Derivative of the Natural Logarithm)
- Proof: (related to Proposition: Derivative of the Reciprocal Function)
- Proof: (related to Proposition: Derivative of the n-th Power Function)
- Proof: (related to Proposition: Derivatives of Even and Odd Functions)
- Proof: (related to Proposition: Difference of Convergent Complex Sequences)
- Proof: (related to Proposition: Difference of Convergent Real Sequences)
- Proof: (related to Proposition: Difference of Convergent Real Series)
- Proof: (related to Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine)
- Proof: (related to Proposition: Differentiable Functions and Tangent-Linear Approximation)
- Proof: (related to Proposition: Differential Equation of the Exponential Function)
- Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Complex Series)
- Proof: (related to Proposition: Direct Comparison Test For Absolutely Convergent Series)
- Proof: (related to Proposition: Direct Comparison Test For Divergent Series)
- Proof: (related to Proposition: Dirichlet's Test)
- Proof: (related to Proposition: Estimate for the Remainder Term of Complex Exponential Function)
- Proof: (related to Proposition: Estimate for the Remainder Term of Exponential Function)
- Proof: (related to Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine)
- Proof: (related to Proposition: Euler's Formula)
- Proof: (related to Proposition: Eveness (Oddness) of Polynomials)
- Proof: (related to Proposition: Eveness of the Cosine of a Real Variable)
- Proof: (related to Proposition: Exponential Function of General Base With Integer Exponents)
- Proof: (related to Proposition: Exponential Function)
- Proof: (related to Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals)
- Proof: (related to Proposition: Functional Equation of the Complex Exponential Function)
- Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base (Revised))
- Proof: (related to Proposition: Functional Equation of the Exponential Function of General Base)
- Proof: (related to Proposition: Functional Equation of the Exponential Function)
- Proof: (related to Proposition: Functional Equation of the Natural Logarithm)
- Proof: (related to Proposition: Gamma Function Interpolates the Factorial)
- Proof: (related to Proposition: Gamma Function)
- Proof: (related to Proposition: General Powers of Positive Numbers)
- Proof: (related to Proposition: Generalized Product Rule)
- Proof: (related to Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members)
- Proof: (related to Proposition: How Convergence Preserves the Order Relation of Sequence Members)
- Proof: (related to Proposition: Hölder's Inequality for Integral p-norms)
- Proof: (related to Proposition: Hölder's Inequality)
- Proof: (related to Proposition: Identity Function is Continuous)
- Proof: (related to Proposition: Image of a Compact Set Under a Continuous Function)
- Proof: (related to Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials)
- Proof: (related to Proposition: Infinite Geometric Series)
- Proof: (related to Proposition: Infinite Series for Cosine and Sine)
- Proof: (related to Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function)
- Proof: (related to Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function)
- Proof: (related to Proposition: Integral Test for Convergence)
- Proof: (related to Proposition: Integral of Cosine)
- Proof: (related to Proposition: Integral of General Powers)
- Proof: (related to Proposition: Integral of Inverse Sine)
- Proof: (related to Proposition: Integral of Sine)
- Proof: (related to Proposition: Integral of the Exponential Function)
- Proof: (related to Proposition: Integral of the Inverse Tangent)
- Proof: (related to Proposition: Integral of the Natural Logarithm)
- Proof: (related to Proposition: Integral of the Reciprocal Function)
- Proof: (related to Proposition: Inverse Tangent and Complex Exponential Function)
- Proof: (related to Proposition: Inverse Tangent of a Real Variable)
- Proof: (related to Proposition: Legendre Polynomials and Legendre Differential Equations)
- Proof: (related to Proposition: Leibniz Criterion for Alternating Series)
- Proof: (related to Proposition: Limit Comparizon Test)
- Proof: (related to Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence)
- Proof: (related to Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence)
- Proof: (related to Proposition: Limit Test for Roots or Ratios)
- Proof: (related to Proposition: Limit of 1/n)
- Proof: (related to Proposition: Limit of Exponential Growth as Compared to Polynomial Growth)
- Proof: (related to Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth)
- Proof: (related to Proposition: Limit of Nested Real Intervals)
- Proof: (related to Proposition: Limit of Nth Root of N)
- Proof: (related to Proposition: Limit of Nth Root of a Positive Constant)
- Proof: (related to Proposition: Limit of a Function is Unique If It Exists)
- Proof: (related to Proposition: Limit of a Rational Function)
- Proof: (related to Proposition: Limit of the Constant Function)
- Proof: (related to Proposition: Limit of the Identity Function)
- Proof: (related to Proposition: Limits of General Powers)
- Proof: (related to Proposition: Limits of Logarithm in `$[0,+\infty]$`)
- Proof: (related to Proposition: Limits of Polynomials at Infinity)
- Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral for Step Functions)
- Proof: (related to Proposition: Linearity and Monotony of the Riemann Integral)
- Proof: (related to Proposition: Logarithm to a General Base)
- Proof: (related to Proposition: Minkowski's Inequality for Integral p-norms)
- Proof: (related to Proposition: Minkowski's Inequality)
- Proof: (related to Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable)
- Proof: (related to Proposition: Monotony Criterion)
- Proof: (related to Proposition: Natural Logarithm)
- Proof: (related to Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers)
- Proof: (related to Proposition: Not all Continuous Functions are also Uniformly Continuous)
- Proof: (related to Proposition: Nth Powers)
- Proof: (related to Proposition: Nth Roots of Positive Numbers)
- Proof: (related to Proposition: Oddness of the Sine of a Real Variable)
- Proof: (related to Proposition: Only the Uniform Convergence Preserves Continuity)
- Proof: (related to Proposition: Open Intervals Contain Uncountably Many Irrational Numbers)
- Proof: (related to Proposition: Open Real Intervals are Uncountable)
- Proof: (related to Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable)
- Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain)
- Proof: (related to Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions)
- Proof: (related to Proposition: Preservation of Inequalities for Limits of Functions)
- Proof: (related to Proposition: Product of Convegent Complex Sequences)
- Proof: (related to Proposition: Product of Convegent Real Sequences)
- Proof: (related to Proposition: Product of Riemann-integrable Functions is Riemann-integrable)
- Proof: (related to Proposition: Product of a Complex Number and a Convergent Complex Sequence)
- Proof: (related to Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity)
- Proof: (related to Proposition: Product of a Real Number and a Convergent Real Sequence)
- Proof: (related to Proposition: Product of a Real Number and a Convergent Real Series)
- Proof: (related to Proposition: Pythagorean Identity)
- Proof: (related to Proposition: Quotient of Convergent Complex Sequences)
- Proof: (related to Proposition: Quotient of Convergent Real Sequences)
- Proof: (related to Proposition: Raabe's Test)
- Proof: (related to Proposition: Rational Functions are Continuous)
- Proof: (related to Proposition: Rational Numbers are Dense in Real Numbers)
- Proof: (related to Proposition: Rational Powers of Positive Numbers)
- Proof: (related to Proposition: Real Sequences Contain Monotonic Subsequences)
- Proof: (related to Proposition: Rearrangement of Absolutely Convergent Series)
- Proof: (related to Proposition: Rearrangement of Convergent Series)
- Proof: (related to Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence)
- Proof: (related to Proposition: Riemann Integral for Step Functions)
- Proof: (related to Proposition: Riemann Sum Converging To the Riemann Integral)
- Proof: (related to Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions)
- Proof: (related to Proposition: Root Test)
- Proof: (related to Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function)
- Proof: (related to Proposition: Square Roots)
- Proof: (related to Proposition: Step Function on Closed Intervals are Riemann-Integrable)
- Proof: (related to Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval)
- Proof: (related to Proposition: Sufficient Condition for a Local Extremum)
- Proof: (related to Proposition: Sum of Arguments of Hyperbolic Cosine)
- Proof: (related to Proposition: Sum of Arguments of Hyperbolic Sine)
- Proof: (related to Proposition: Sum of Convergent Complex Sequences)
- Proof: (related to Proposition: Sum of Convergent Real Sequences)
- Proof: (related to Proposition: Sum of Convergent Real Series)
- Proof: (related to Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty)
- Proof: (related to Proposition: Supremum Norm and Uniform Convergence)
- Proof: (related to Proposition: Taylor's Formula with Remainder Term of Lagrange)
- Proof: (related to Proposition: The distance of complex numbers makes complex numbers a metric space.)
- Proof: (related to Proposition: The distance of real numbers makes real numbers a metric space.)
- Proof: (related to Proposition: Uniform Convergence Criterion of Cauchy)
- Proof: (related to Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series)
- Proof: (related to Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions)
- Proof: (related to Proposition: Uniqueness Of the Limit of a Sequence)
- Proof: (related to Proposition: Zero of Cosine)
- Proof: (related to Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum)
- Proof: (related to Proposition: `\(\exp(0)=1\)` (Complex Case))
- Proof: (related to Proposition: `\(\exp(0)=1\)`)
- Proof: (related to Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences)
- Proof: (related to Proposition: n-th Roots of Unity)
- Proof: (related to Theorem: Completeness Principle for Complex Numbers)
- Proof: (related to Theorem: Completeness Principle for Real Numbers)
- Proof: (related to Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains)
- Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
- Proof: (related to Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous)
- Proof: (related to Theorem: Darboux's Theorem)
- Proof: (related to Theorem: Defining Properties of the Field of Real Numbers)
- Proof: (related to Theorem: Every Bounded Monotonic Sequence Is Convergent)
- Proof: (related to Theorem: Fundamental Theorem of Calculus)
- Proof: (related to Theorem: Heine-Borel Theorem)
- Proof: (related to Theorem: Indefinite Integral, Antiderivative)
- Proof: (related to Theorem: Inequality of Weighted Arithmetic Mean)
- Proof: (related to Theorem: Inequality of the Arithmetic Mean)
- Proof: (related to Theorem: Integration by Substitution)
- Proof: (related to Theorem: Intermediate Value Theorem)
- Proof: (related to Theorem: Mean Value Theorem For Riemann Integrals)
- Proof: (related to Theorem: Nested Closed Subset Theorem)
- Proof: (related to Theorem: Partial Integration)
- Proof: (related to Theorem: Reverse Triangle Inequalities)
- Proof: (related to Theorem: Rolle's Theorem)
- Proof: (related to Theorem: Squeezing Theorem for Functions)
- Proof: (related to Theorem: Supremum Property, Infimum Property)
- Proof: (related to Theorem: Triangle Inequality)
- Proof: (related to Corollary: Reciprocity Law of Falling And Rising Factorial Powers)
- Proof: (related to Lemma: Stirling Numbers and Rising Factorial Powers)
- Proof: (related to Proposition: Antidifferences are Unique Up to a Periodic Constant)
- Proof: (related to Proposition: Antidifferences of Some Functions)
- Proof: (related to Proposition: Basic Calculations Involving Indefinite Sums)
- Proof: (related to Proposition: Basic Calculations Involving the Difference Operator)
- Proof: (related to Proposition: Closed Formula For Binomial Coefficients)
- Proof: (related to Proposition: Comparison between the Stirling numbers of the First and Second Kind)
- Proof: (related to Proposition: Difference Operator of Falling Factorial Powers)
- Proof: (related to Proposition: Difference Operator of Powers)
- Proof: (related to Proposition: Factorial Polynomials have a Unique Representation)
- Proof: (related to Proposition: Factorial)
- Proof: (related to Proposition: Factorials and Stirling Numbers of the First Kind)
- Proof: (related to Proposition: Fundamental Counting Principle)
- Proof: (related to Proposition: Indicator Function and Set Operations)
- Proof: (related to Proposition: Inversion Formulas For Stirling Numbers)
- Proof: (related to Proposition: Multinomial Coefficient)
- Proof: (related to Proposition: Number of Ordered n-Tuples in a Set)
- Proof: (related to Proposition: Number of Relations on a Finite Set)
- Proof: (related to Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters)
- Proof: (related to Proposition: Number of Subsets of a Finite Set)
- Proof: (related to Proposition: Recursive Formula for Binomial Coefficients)
- Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the First Kind)
- Proof: (related to Proposition: Recursive Formula for the Stirling Numbers of the Second Kind)
- Proof: (related to Proposition: Recursively Defined Arithmetic Functions, Recursion)
- Proof: (related to Proposition: Simple Binomial Identities)
- Proof: (related to Theorem: Approximation of Factorials Using the Stirling Formula)
- Proof: (related to Theorem: Fundamental Theorem of the Difference Calculus)
- Proof: (related to Theorem: Inclusion-Exclusion Principle (Sylvester's Formula))
- Proof: (related to Theorem: Taylor's Formula Using the Difference Operator)
- Proof: (related to Lemma: Equivalence of Different Descriptions of a Straight Line Using Two Vectors)
- Proof: (related to Proposition: Presentation of a Straight Line in a Plane as a Linear Equation)
- Proof: (related to Theorem: Existence and Uniqueness of a Straight Line Through Two Points)
- Proof: (related to Proposition: Common Points of Two Distinct Straight Lines in a Plane)
- Proof: (related to Proposition: Common Points of a Plane and a Straight Line Not in the Plane)
- Proof: (related to Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line)
- Proof: (related to Proposition: Plane Determined by two Crossing Straight Lines)
- Proof: (related to Algorithm: Get All Components of a Graph)
- Proof: (related to Algorithm: Get All Components of a Graph)
- Proof: (related to Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex)
- Proof: (related to Algorithm: Get the Cut Vertices and Biconnected Components of a Connected Graph)
- Proof: (related to Corollary: Bounds for the Minimal Tree Decomposability)
- Proof: (related to Corollary: Even Number of Vertices with an Odd Degree in Finite Digraphs)
- Proof: (related to Corollary: Even Number of Vertices with an Odd Degree in Finite Graphs)
- Proof: (related to Corollary: Number of Vertices, Edges, and Faces of a Dual Graph)
- Proof: (related to Corollary: Planarity of Subdivisions)
- Proof: (related to Lemma: Biconnectivity is a Necessary Condition for a Hamiltonian Graph)
- Proof: (related to Lemma: Coloring of Trees)
- Proof: (related to Lemma: Dual Graph of a All Faces Contained in a Planar Hamiltonian Cycle is a Tree)
- Proof: (related to Lemma: Handshaking Lemma for Finite Digraphs)
- Proof: (related to Lemma: Handshaking Lemma for Finite Graphs)
- Proof: (related to Lemma: Handshaking Lemma for Planar Graphs)
- Proof: (related to Lemma: Lower Bound of Leaves in a Tree)
- Proof: (related to Lemma: Size of an `\(r\)`-regular Graph with `\(n\)` Vertices)
- Proof: (related to Lemma: Splitting a Graph with Even Degree Vertices into Cycles)
- Proof: (related to Lemma: When is it possible to find a separating cycle in a biconnected graph, given a non-separating cycle?)
- Proof: (related to Proposition: A Necessary Condition for a Graph to be Planar)
- Proof: (related to Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II))
- Proof: (related to Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar)
- Proof: (related to Proposition: Characterization of Cutvertices)
- Proof: (related to Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph)
- Proof: (related to Proposition: Relationship Between Planarity and Biconnectivity of Graphs)
- Proof: (related to Proposition: Relationship Between Planarity and Connectivity of Graphs)
- Proof: (related to Theorem: Brooks' Theorem)
- Proof: (related to Theorem: Characterization of Biconnected Planar Graphs)
- Proof: (related to Theorem: Characterization of Bipartite Graphs)
- Proof: (related to Theorem: Characterization of Eulerian Graphs)
- Proof: (related to Theorem: Characterization of Planar Graphs)
- Proof: (related to Theorem: Characterization of Planar Hamiltonian Graphs)
- Proof: (related to Theorem: Characterization of Semi-Eulerian Graphs)
- Proof: (related to Theorem: Euler Characteristic for Planar Graphs)
- Proof: (related to Theorem: Five Color Theorem for Planar Graphs)
- Proof: (related to Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph)
- Proof: (related to Theorem: Four Color Theorem for Planar Graphs)
- Proof: (related to Theorem: Number of Labeled Spanning Trees)
- Proof: (related to Theorem: Six Color Theorem for Planar Graphs)
- Proof: (related to Proposition: Equivalent Knot Diagrams)
- Proof: (related to Corollary: Algebraic Structure of Strings over an Alphabet)
- Proof: (related to Corollary: All Boolean Functions Can Be Built Using Conjunction, Disjunction, and Negation)
- Proof: (related to Corollary: Commutativity of Conjunction)
- Proof: (related to Corollary: Commutativity of Disjunction)
- Proof: (related to Corollary: Commutativity of Equivalence)
- Proof: (related to Lemma: A Criterion for Valid Logical Arguments)
- Proof: (related to Lemma: A proposition cannot be both, true and false)
- Proof: (related to Lemma: A proposition cannot be equivalent to its negation)
- Proof: (related to Lemma: Affirming the Consequent of an Implication)
- Proof: (related to Lemma: Boolean Algebra of Propositional Logic)
- Proof: (related to Lemma: Boolean Function)
- Proof: (related to Lemma: Construction of Conjunctive and Disjunctive Canonical Normal Forms)
- Proof: (related to Lemma: De Morgan's Laws (Logic))
- Proof: (related to Lemma: Denying the Antecedent of an Implication)
- Proof: (related to Lemma: Disjunctive Syllogism)
- Proof: (related to Lemma: Distributivity of Conjunction and Disjunction)
- Proof: (related to Lemma: Every Contraposition to a Proposition is a Tautology to this Proposition)
- Proof: (related to Lemma: Every Proposition Implies Itself)
- Proof: (related to Lemma: Hypothetical Syllogism)
- Proof: (related to Lemma: Implication as a Disjunction)
- Proof: (related to Lemma: It is true that something can be (either) true or false)
- Proof: (related to Lemma: Mixing-up the Inclusive and Exclusive Disjunction)
- Proof: (related to Lemma: Mixing-up the Sufficient and Necessary Conditions)
- Proof: (related to Lemma: Modus Ponens)
- Proof: (related to Lemma: Modus Tollens)
- Proof: (related to Lemma: Negation of an Implication)
- Proof: (related to Lemma: The Proving Principle By Contraposition, Contrapositive)
- Proof: (related to Lemma: The Proving Principle by Complete Induction)
- Proof: (related to Lemma: The Proving Principle by Contradiction)
- Proof: (related to Lemma: The Proving Principle by Transfinite Induction)
- Proof: (related to Lemma: Unique Valuation of Minterms and Maxterms)
- Proof: (related to Proposition: Associativity of Conjunction)
- Proof: (related to Proposition: Associativity of Disjunction)
- Proof: (related to Corollary: A product of two real numbers is zero if and only if at least one of these numbers is zero.)
- Proof: (related to Corollary: Contraposition of Cancellative Law for Adding Natural Numbers)
- Proof: (related to Corollary: Existence of Arbitrarily Small Positive Rational Numbers)
- Proof: (related to Corollary: Existence of Arbitrarily Small Powers)
- Proof: (related to Corollary: Existence of Natural Numbers Exceeding Positive Real Numbers (Archimedian Principle))
- Proof: (related to Corollary: Existence of Natural One (Neutral Element of Multiplication of Natural Numbers))
- Proof: (related to Corollary: Existence of Natural Zero (Neutral Element of Addition of Natural Numbers))
- Proof: (related to Corollary: Existence of Powers Exceeding Any Positive Constant)
- Proof: (related to Corollary: Existence of Unique Integers Exceeding Real Numbers)
- Proof: (related to Corollary: General Associative Law of Multiplication)
- Proof: (related to Corollary: General Commutative Law of Multiplication)
- Proof: (related to Corollary: Order Relation for Natural Numbers is Strict Total)
- Proof: (related to Corollary: Properties of the Absolute Value)
- Proof: (related to Corollary: Rules of Calculations with Inequalities)
- Proof: (related to Corollary: The absolute value makes the set of rational numbers a metric space.)
- Proof: (related to Corollary: `\((-x)(-y)=xy\)`)
- Proof: (related to Corollary: `\((-x)y=-(xy)\)`)
- Proof: (related to Corollary: `\((x^{-1})^{-1}=x\)`)
- Proof: (related to Corollary: `\(-(-x)=x\)`)
- Proof: (related to Corollary: `\(-0=0\)`)
- Proof: (related to Corollary: `\(0x=0\)`)
- Proof: (related to Corollary: `\(1^{-1}=1\)`)
- Proof: (related to Lemma: Complex Numbers are Two-Dimensional and the Complex Numbers `\(1\)` and Imaginary Unit `\(i\)` Form Their Basis)
- Proof: (related to Lemma: Convergent Rational Sequences With Limit `\(0\)` Are Rational Cauchy Sequences)
- Proof: (related to Lemma: Convergent Rational Sequences With Limit `\(0\)` Are a Subgroup of Rational Cauchy Sequences With Respect To Addition)
- Proof: (related to Lemma: Convergent Rational Sequences With Limit `\(0\)` Are an Ideal Of the Ring of Rational Cauchy Sequences)
- Proof: (related to Lemma: Linear Independence of the Imaginary Unit `\(i\)` and the Complex Number `\(1\)`)
- Proof: (related to Lemma: Rational Cauchy Sequences Build a Commutative Group With Respect To Addition)
- Proof: (related to Lemma: Rational Cauchy Sequences Build a Commutative Monoid With Respect To Multiplication)
- Proof: (related to Lemma: Unit Ring of All Rational Cauchy Sequences)
- Proof: (related to Proposition: Abelian Partial Summation Method)
- Proof: (related to Proposition: Absolute Value of Complex Conjugate)
- Proof: (related to Proposition: Absolute Value of the Product of Complex Numbers)
- Proof: (related to Proposition: Addition Of Natural Numbers)
- Proof: (related to Proposition: Addition Of Rational Numbers)
- Proof: (related to Proposition: Addition Of Real Numbers Is Associative)
- Proof: (related to Proposition: Addition Of Real Numbers Is Commutative)
- Proof: (related to Proposition: Addition of Complex Numbers Is Associative)
- Proof: (related to Proposition: Addition of Complex Numbers Is Commutative)
- Proof: (related to Proposition: Addition of Integers Is Associative)
- Proof: (related to Proposition: Addition of Integers Is Associative)
- Proof: (related to Proposition: Addition of Integers Is Cancellative)
- Proof: (related to Proposition: Addition of Integers Is Commutative)
- Proof: (related to Proposition: Addition of Integers)
- Proof: (related to Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities)
- Proof: (related to Proposition: Addition of Rational Cauchy Sequences Is Associative)
- Proof: (related to Proposition: Addition of Rational Cauchy Sequences Is Cancellative)
- Proof: (related to Proposition: Addition of Rational Cauchy Sequences Is Commutative)
- Proof: (related to Proposition: Addition of Rational Cauchy Sequences)
- Proof: (related to Proposition: Addition of Rational Numbers Is Associative)
- Proof: (related to Proposition: Addition of Rational Numbers Is Cancellative)
- Proof: (related to Proposition: Addition of Rational Numbers Is Commutative)
- Proof: (related to Proposition: Addition of Real Numbers Is Cancellative)
- Proof: (related to Proposition: Addition of Real Numbers)
- Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Addition)
- Proof: (related to Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Complex Numbers Together with Addition and Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Complex Numbers Together with Addition)
- Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition and Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
- Proof: (related to Proposition: Algebraic Structure of Integers Together with Addition)
- Proof: (related to Proposition: Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Rational Numbers Together with Addition)
- Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication)
- Proof: (related to Proposition: Algebraic Structure of Real Numbers Together with Addition)
- Proof: (related to Proposition: Alternating Sum of Binomial Coefficients)
- Proof: (related to Proposition: Basic Rules of Manipulating Finite Sums)
- Proof: (related to Proposition: Calculating with Complex Conjugates)
- Proof: (related to Proposition: Comparing Natural Numbers Using the Concept of Addition)
- Proof: (related to Proposition: Complex Numbers Cannot Be Ordered)
- Proof: (related to Proposition: Complex Numbers are a Field Extension of Real Numbers)
- Proof: (related to Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers)
- Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Integers)
- Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Rational Numbers)
- Proof: (related to Proposition: Contraposition of Cancellative Law for Adding Real Numbers)
- Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Integers)
- Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers)
- Proof: (related to Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers)
- Proof: (related to Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers)
- Proof: (related to Proposition: Definition of Integers)
- Proof: (related to Proposition: Definition of Rational Numbers)
- Proof: (related to Proposition: Definition of Real Numbers)
- Proof: (related to Proposition: Discovery of Irrational Numbers)
- Proof: (related to Proposition: Discovery of Irrational Numbers)
- Proof: (related to Proposition: Distributivity Law For Integers)
- Proof: (related to Proposition: Distributivity Law For Rational Cauchy Sequences)
- Proof: (related to Proposition: Distributivity Law For Rational Numbers)
- Proof: (related to Proposition: Distributivity Law For Real Numbers)
- Proof: (related to Proposition: Distributivity Law for Complex Numbers)
- Proof: (related to Proposition: Double Summation)
- Proof: (related to Proposition: Equality of Two Ratios)
- Proof: (related to Proposition: Every Natural Number Is Greater or Equal Zero)
- Proof: (related to Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers)
- Proof: (related to Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers))
- Proof: (related to Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers))
- Proof: (related to Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers))
- Proof: (related to Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers))
- Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Addition)
- Proof: (related to Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication)
- Proof: (related to Proposition: Existence of Inverse Integers With Respect to Addition)
- Proof: (related to Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition)
- Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Addition)
- Proof: (related to Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication)
- Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Addition)
- Proof: (related to Proposition: Existence of Inverse Real Numbers With Respect to Multiplication)
- Proof: (related to Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences))
- Proof: (related to Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences))
- Proof: (related to Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers))
- Proof: (related to Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers))
- Proof: (related to Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers))
- Proof: (related to Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers))
- Proof: (related to Proposition: Extracting the Real and the Imaginary Part of a Complex Number)
- Proof: (related to Proposition: Imaginary Unit)
- Proof: (related to Proposition: Inequality of Natural Numbers and Their Successors)
- Proof: (related to Proposition: Multiplication Of Rational Cauchy Sequences)
- Proof: (related to Proposition: Multiplication Of Rational Numbers Is Cancellative)
- Proof: (related to Proposition: Multiplication Of Rational Numbers Is Commutative)
- Proof: (related to Proposition: Multiplication Of Rational Numbers)
- Proof: (related to Proposition: Multiplication of Complex Numbers Is Associative)
- Proof: (related to Proposition: Multiplication of Complex Numbers Is Commutative)
- Proof: (related to Proposition: Multiplication of Complex Numbers Using Polar Coordinates)
- Proof: (related to Proposition: Multiplication of Integers Is Associative)
- Proof: (related to Proposition: Multiplication of Integers Is Cancellative)
- Proof: (related to Proposition: Multiplication of Integers Is Commutative)
- Proof: (related to Proposition: Multiplication of Integers)
- Proof: (related to Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation)
- Proof: (related to Proposition: Multiplication of Rational Cauchy Sequences Is Associative)
- Proof: (related to Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative)
- Proof: (related to Proposition: Multiplication of Rational Cauchy Sequences Is Commutative)
- Proof: (related to Proposition: Multiplication of Rational Numbers Is Associative)
- Proof: (related to Proposition: Multiplication of Real Numbers Is Associative)
- Proof: (related to Proposition: Multiplication of Real Numbers Is Cancellative)
- Proof: (related to Proposition: Multiplication of Real Numbers Is Commutative)
- Proof: (related to Proposition: Multiplication of Real Numbers)
- Proof: (related to Proposition: Multiplying Negative and Positive Integers)
- Proof: (related to Proposition: Multiplying Negative and Positive Rational Numbers)
- Proof: (related to Proposition: Multiplying Negative and Positive Real Numbers)
- Proof: (related to Proposition: Order Relation for Integers is Strict Total)
- Proof: (related to Proposition: Order Relation for Natural Numbers, Revised)
- Proof: (related to Proposition: Order Relation for Rational Numbers is Strict Total)
- Proof: (related to Proposition: Order Relation for Real Numbers is Strict and Total)
- Proof: (related to Proposition: Polar Coordinates of a Complex Number)
- Proof: (related to Proposition: Position of Minus Sign in Rational Numbers Representations)
- Proof: (related to Proposition: Product of Two Ratios)
- Proof: (related to Proposition: Product of Two Sums (Generalized Distributivity Rule))
- Proof: (related to Proposition: Ratio of Two Ratios)
- Proof: (related to Proposition: Rational Cauchy Sequence Members Are Bounded)
- Proof: (related to Proposition: Rule of Combining Different Sets of Indices)
- Proof: (related to Proposition: Sum and Difference of Two Ratios)
- Proof: (related to Proposition: Sum of Arithmetic Progression)
- Proof: (related to Proposition: Sum of Binomial Coefficients I)
- Proof: (related to Proposition: Sum of Binomial Coefficients II)
- Proof: (related to Proposition: Sum of Binomial Coefficients III)
- Proof: (related to Proposition: Sum of Binomial Coefficients IV)
- Proof: (related to Proposition: Sum of Binomial Coefficients)
- Proof: (related to Proposition: Sum of Cosines)
- Proof: (related to Proposition: Sum of Cube Numbers)
- Proof: (related to Proposition: Sum of Geometric Progression)
- Proof: (related to Proposition: Sum of Squares)
- Proof: (related to Proposition: The General Perturbation Method)
- Proof: (related to Proposition: Transitivity of the Order Relation of Natural Numbers)
- Proof: (related to Proposition: Unique Solvability of `$ax=b$`)
- Proof: (related to Proposition: Unique Solvability of `\(a+x=b\)`)
- Proof: (related to Proposition: Uniqueness Of Natural One)
- Proof: (related to Proposition: Uniqueness Of Predecessors Of Natural Numbers)
- Proof: (related to Proposition: Uniqueness Of Rational One)
- Proof: (related to Proposition: Uniqueness of Complex Zero)
- Proof: (related to Proposition: Uniqueness of Integer One)
- Proof: (related to Proposition: Uniqueness of Integer Zero)
- Proof: (related to Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication)
- Proof: (related to Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication)
- Proof: (related to Proposition: Uniqueness of Natural Zero)
- Proof: (related to Proposition: Uniqueness of Negative Numbers)
- Proof: (related to Proposition: Uniqueness of Rational Zero)
- Proof: (related to Proposition: Uniqueness of Real One)
- Proof: (related to Proposition: Uniqueness of Real Zero)
- Proof: (related to Proposition: Well-Ordering Principle of Natural Numbers)
- Proof: (related to Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)`)
- Proof: (related to Proposition: `\(-(x+y)=-x-y\)`)
- Proof: (related to Corollary: Diophantine Equations of Congruences Have a Finite Number Of Solutions)
- Proof: (related to Corollary: Divisors of a Product Of Many Factors, Co-Prime to All But One Factor, Divide This Factor)
- Proof: (related to Corollary: Primality of the Smallest Non-Trivial Divisor)
- Proof: (related to Corollary: Prime Dividing Product of Primes Implies Prime Divisor)
- Proof: (related to Corollary: Simple Conclusions For Multiplicative Functions)
- Proof: (related to Corollary: Sums, Products, and Powers Of Congruences)
- Proof: (related to Lemma: Coprimality and Congruence Classes)
- Proof: (related to Lemma: Division with Quotient and Remainder)
- Proof: (related to Lemma: Gaussian Lemma (Number Theory))
- Proof: (related to Lemma: Generalized Euclidean Lemma)
- Proof: (related to Lemma: Möbius and Floor Functions Combined)
- Proof: (related to Lemma: Reciprocity Law for Floor Functions)
- Proof: (related to Lemma: Sets of Integers Co-Prime to a given Integer are Divisor-Closed)
- Proof: (related to Lemma: Sums of Floors)
- Proof: (related to Lemma: Upper Bound of Harmonic Series Times Möbius Function)
- Proof: (related to Proposition: A Linear Term for 1 Using Two Co-prime Coefficients)
- Proof: (related to Proposition: A Necessary Condition for an Integer to be Prime)
- Proof: (related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`)
- Proof: (related to Proposition: All Solutions Given a Solution of an LDE With Two Variables)
- Proof: (related to Proposition: Calculating the Number of Positive Divisors)
- Proof: (related to Proposition: Calculating the Sum of Divisors)
- Proof: (related to Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$`)
- Proof: (related to Proposition: Cancellation of Congruences with General Factor)
- Proof: (related to Proposition: Co-prime Primes)
- Proof: (related to Proposition: Complete and Reduced Residue Systems (Revised))
- Proof: (related to Proposition: Congruence Classes)
- Proof: (related to Proposition: Congruence Modulo a Divisor)
- Proof: (related to Proposition: Congruences and Division with Quotient and Remainder)
- Proof: (related to Proposition: Connection between Quotient, Remainder, Modulo and Floor Function)
- Proof: (related to Proposition: Convergence of Alternating Harmonic Series)
- Proof: (related to Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number)
- Proof: (related to Proposition: Counting the Solutions of Diophantine Equations of Congruences)
- Proof: (related to Proposition: Creation of Complete Residue Systems From Others)
- Proof: (related to Proposition: Creation of Reduced Residue Systems From Others)
- Proof: (related to Proposition: Diophantine Equations of Congruences)
- Proof: (related to Proposition: Divergence of Harmonic Series)
- Proof: (related to Proposition: Divisibility Laws)
- Proof: (related to Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor)
- Proof: (related to Proposition: Euler's Criterion For Quadratic Residues)
- Proof: (related to Proposition: Even Perfect Numbers)
- Proof: (related to Proposition: Every Integer Is Either Even or Odd)
- Proof: (related to Proposition: Existence of Prime Divisors)
- Proof: (related to Proposition: Factorization of Greatest Common Divisor and Least Common Multiple)
- Proof: (related to Proposition: Finite Number of Divisors)
- Proof: (related to Proposition: Floor Function and Division with Quotient and Remainder)
- Proof: (related to Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor)
- Proof: (related to Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers)
- Proof: (related to Proposition: Greatest Common Divisor of More Than Two Numbers)
- Proof: (related to Proposition: Greatest Common Divisors Of Integers and Prime Numbers)
- Proof: (related to Proposition: Least Common Multiple of More Than Two Numbers)
- Proof: (related to Proposition: Least Common Multiple)
- Proof: (related to Proposition: Legendre Symbols of Equal Residues)
- Proof: (related to Proposition: Multiplication of Congruences with a Positive Factor)
- Proof: (related to Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$`)
- Proof: (related to Proposition: Multiplicativity of the Legendre Symbol)
- Proof: (related to Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors)
- Proof: (related to Proposition: Number of Multiples of a Given Number Less Than Another Number)
- Proof: (related to Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime)
- Proof: (related to Proposition: Numbers Being the Product of Their Divisors)
- Proof: (related to Proposition: Product of Two Even Numbers)
- Proof: (related to Proposition: Product of Two Odd Numbers)
- Proof: (related to Proposition: Product of an Even and an Odd Number)
- Proof: (related to Proposition: Properties of Floors and Ceilings)
- Proof: (related to Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple)
- Proof: (related to Proposition: Sign of Divisors of Integers)
- Proof: (related to Proposition: Sum of Möbius Function Over Divisors)
- Proof: (related to Theorem: Commutative Group of Multiplicative Functions)
- Proof: (related to Theorem: Euler-Fermat Theorem)
- Proof: (related to Theorem: Fermat's Last Theorem)
- Proof: (related to Theorem: First Supplementary Law to the Quadratic Reciprocity Law)
- Proof: (related to Theorem: Fundamental Theorem of Arithmetic)
- Proof: (related to Theorem: Infinite Set of Prime Numbers)
- Proof: (related to Theorem: Infinite Set of Prime Numbers)
- Proof: (related to Theorem: Infinite Set of Prime Numbers)
- Proof: (related to Theorem: Möbius Inversion Formula)
- Proof: (related to Theorem: Number of Multiples of a Prime Number Less Than Factorial)
- Proof: (related to Theorem: Properties of the Jacobi Symbol)
- Proof: (related to Theorem: Quadratic Reciprocity Law)
- Proof: (related to Theorem: Second Supplementary Law to the Quadratic Reciprocity Law)
- Proof: (related to Lemma: Sieve for Twin Primes)
- Proof: (related to Corollary: Probability of Laplace Experiments)
- Proof: (related to Corollary: Probability of the Impossible Event)
- Proof: (related to Proposition: Binomial Distribution)
- Proof: (related to Proposition: Characterization of Independent Events II)
- Proof: (related to Proposition: Characterization of Independent Events)
- Proof: (related to Proposition: Geometric Distribution)
- Proof: (related to Proposition: Monotonically Increasing Property of Probability Distributions)
- Proof: (related to Proposition: Multinomial Distribution)
- Proof: (related to Proposition: Probability of Event Difference)
- Proof: (related to Proposition: Probability of Event Union)
- Proof: (related to Proposition: Probability of Included Event)
- Proof: (related to Proposition: Probability of Joint Events)
- Proof: (related to Proposition: Probability of the Complement Event)
- Proof: (related to Proposition: Replacing Mutually Independent Events by Their Complements)
- Proof: (related to Proposition: Urn Model With Replacement)
- Proof: (related to Proposition: Urn Model Without Replacement)
- Proof: (related to Theorem: Bayes' Theorem)
- Proof: (related to Theorem: Law of Total Probability)
- Proof: (related to Theorem: Theorem of Large Numbers for Relative Frequencies)
- Proof: (related to Corollary: Equality of Sets)
- Proof: (related to Corollary: Irrational Numbers are Uncountable)
- Proof: (related to Corollary: Justification of Power Set)
- Proof: (related to Corollary: Justification of Set Intersection)
- Proof: (related to Corollary: Justification of Set Union)
- Proof: (related to Corollary: Justification of Subsets and Supersets)
- Proof: (related to Corollary: Justification of the Difference)
- Proof: (related to Corollary: Justification of the Set-Builder Notation)
- Proof: (related to Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets)
- Proof: (related to Corollary: Properties of Transitive Sets)
- Proof: (related to Corollary: Set Difference and Set Complement are the Same Concepts)
- Proof: (related to Corollary: Strictly, Well-ordered Sets and Transitive Sets)
- Proof: (related to Corollary: There is no set of all sets)
- Proof: (related to Corollary: Uniqueness of the Empty Set)
- Proof: (related to Lemma: Any Set is Subset of Some Transitive Set - Its Transitive Hull)
- Proof: (related to Lemma: Behavior of Functions with Set Operations)
- Proof: (related to Lemma: Comparing the Elements of Strictly Ordered Sets)
- Proof: (related to Lemma: Composition of Functions)
- Proof: (related to Lemma: Composition of Relations (Sometimes) Preserves Their Left-Total Property)
- Proof: (related to Lemma: Composition of Relations Preserves Their Right-Uniqueness Property)
- Proof: (related to Lemma: Equivalence of Set Inclusion and Element Inclusion of Ordinals)
- Proof: (related to Lemma: Finite Cardinal Numbers and Set Operations)
- Proof: (related to Lemma: Properties of Ordinal Numbers)
- Proof: (related to Lemma: Successor of Ordinal)
- Proof: (related to Lemma: Zorn's Lemma)
- Proof: (related to Proposition: Cardinal Number)
- Proof: (related to Proposition: Cardinals of a Set and Its Power Set)
- Proof: (related to Proposition: Characterization of Bijective Functions)
- Proof: (related to Proposition: Composition of Bijective Functions is Bijective)
- Proof: (related to Proposition: Composition of Functions is Associative)
- Proof: (related to Proposition: Composition of Injective Functions is Injective)
- Proof: (related to Proposition: Composition of Surjective Functions is Surjective)
- Proof: (related to Proposition: Contained Relation is a Strict Order)
- Proof: (related to Proposition: Counting the Set's Elements Using Its Partition)
- Proof: (related to Proposition: De Morgan's Laws (Sets))
- Proof: (related to Proposition: Distributivity Laws For Sets)
- Proof: (related to Proposition: Equivalent Notions of Ordinals)
- Proof: (related to Proposition: Finite Chains are Well-ordered)
- Proof: (related to Proposition: Functions Constitute Equivalence Relations)
- Proof: (related to Proposition: Injective, Surjective and Bijective Compositions)
- Proof: (related to Proposition: Intersection of a Set With Another Set is Subset of This Set)
- Proof: (related to Proposition: More Characterizations of Finite Sets)
- Proof: (related to Proposition: Ordinals Are Downward Closed)
- Proof: (related to Proposition: Set Intersection is Associative)
- Proof: (related to Proposition: Set Intersection is Commutative)
- Proof: (related to Proposition: Set Union is Associative)
- Proof: (related to Proposition: Set Union is Commutative)
- Proof: (related to Proposition: Sets and Their Complements)
- Proof: (related to Proposition: Sets are Subsets of Their Union)
- Proof: (related to Proposition: Subset of a Countable Set is Countable)
- Proof: (related to Proposition: Subsets of Finite Sets)
- Proof: (related to Proposition: The Contained Relation is Extensional)
- Proof: (related to Proposition: The Equality of Sets Is an Equivalence Relation)
- Proof: (related to Proposition: The Inverse Of a Composition)
- Proof: (related to Proposition: Transitive Recursion)
- Proof: (related to Proposition: Uncountable and Countable Subsets of Natural Numbers)
- Proof: (related to Proposition: Union of Countably Many Countable Sets)
- Proof: (related to Proposition: Well-ordered Sets are Chains)
- Proof: (related to Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice)
- Proof: (related to Theorem: Distinction Between Finite and Infinite Sets Using Subsets)
- Proof: (related to Theorem: Mostowski's Theorem)
- Proof: (related to Theorem: Schröder-Bernstein Theorem)
- Proof: (related to Theorem: Trichotomy of Ordinals)
- Proof: (related to Proposition: Calculation Rules for the Big O Notation)
- Proof: (related to Corollary: The set of WHILE-computable functions is included in the set of partially WHILE-computable functions)
- Proof: (related to Lemma: LOOP-Computable Functions are Total)
- Proof: (related to Theorem: Simulating LOOP Programs Using WHILE Programs)
- Proof: (related to Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa))
- Proof: (related to Corollary: Reduction of `$\epsilon$`-NFA to DFA)
- Proof: (related to Theorem: Deterministic Finite Automata Describe Regular Languages)
- Proof: (related to Theorem: Reduction of NFA to DFA (Rabin-Scott Theorem))
- Proof: (related to Theorem: Reduction of `$\epsilon$`-NFA to NFA)
- Proof: (related to Algorithm: Calculation of Inverses Modulo a Number (Python))
- Proof: (related to Algorithm: Continued Fraction (Python))
- Proof: (related to Algorithm: Extended Greatest Common Divisor (Python))
- Proof: (related to Algorithm: Greatest Common Divisor (Python))
- Proof: (related to Algorithm: Horner Scheme)
- Proof: (related to Algorithm: Horner Scheme)
- Proof: (related to Algorithm: Jacobi Symbol (Python))
- Proof: (related to Theorem: First Law of Planetary Motion)
- Proof: (related to Theorem: Second Law of Planetary Motion)
- Proof: (related to Theorem: Third Law of Planetary Motion)
- Proof: (related to Proposition: Construction of a Light Clock)
- Proof: (related to Proposition: Time Dilation, Lorentz Factor)
- Proof: (related to Corollary: Every Distance Is Positive Definite)
- Proof: (related to Corollary: Every uniformly convergent sequence of functions is pointwise convergent.)
- Proof: (related to Lemma: Characterization of Closed Sets by Limits of Sequences)
- Proof: (related to Lemma: Convergent Sequences are Cauchy Sequences (Metric Spaces))
- Proof: (related to Lemma: Criteria for Convergent Sequences)
- Proof: (related to Proposition: A Necessary Condition of a Neighborhood to be Open)
- Proof: (related to Proposition: Alternative Characterization of Topological Spaces)
- Proof: (related to Proposition: Bijective Open Functions)
- Proof: (related to Proposition: Characterization of `$T_1$` Spaces)
- Proof: (related to Proposition: Characterization of `$T_2$` Spaces)
- Proof: (related to Proposition: Clopen Sets and Boundaries)
- Proof: (related to Proposition: Construction of Topological Spaces Using a Subbasis)
- Proof: (related to Proposition: Continuity of Compositions of Functions)
- Proof: (related to Proposition: Distance in Normed Vector Spaces)
- Proof: (related to Proposition: Equivalent Notions of Continuous Functions)
- Proof: (related to Proposition: Equivalent Notions of Homeomorphisms)
- Proof: (related to Proposition: Filter Base)
- Proof: (related to Proposition: How the Boundary Changes the Property of a Set of Being Open)
- Proof: (related to Proposition: Inheritance of the `$T_1$` Property)
- Proof: (related to Proposition: Inheritance of the `$T_2$` Property)
- Proof: (related to Proposition: Integral p-Norm)
- Proof: (related to Proposition: Isometry is Injective)
- Proof: (related to Proposition: Maximum Norm as a Limit of p-Norms)
- Proof: (related to Proposition: Metric Spaces and Empty Sets are Clopen)
- Proof: (related to Proposition: Metric Spaces are Hausdorff Spaces)
- Proof: (related to Proposition: Modulus of Continuity is Continuous)
- Proof: (related to Proposition: Modulus of Continuity is Monotonically Increasing)
- Proof: (related to Proposition: Modulus of Continuity is Subadditive)
- Proof: (related to Proposition: Perfect Sets vs. Derived Sets)
- Proof: (related to Proposition: Properties of the Set of All Neighborhoods of a Point)
- Proof: (related to Proposition: Uniqueness of the Limit of a Sequence)
- Proof: (related to Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity)
- Proof: (related to Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm)
- Proof: (related to Theorem: Continuous Functions on Compact Domains are Uniformly Continuous)
- Proof: (related to Theorem: Theorem of Bolzano-Weierstrass)
- Proof: (Euclid) (related to Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line)
- Proof: (Euclid) (related to Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line)
- Proof: (Euclid) (related to Proposition: 1.18: Angles and Sides in a Triangle I)
- Proof: (Euclid) (related to Proposition: 1.19: Angles and Sides in a Triangle II)
- Proof: (Euclid) (related to Proposition: 1.24: Angles and Sides in a Triangle III)
- Proof: (Euclid) (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Proof: (Euclid) (related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)
- Proof: (Euclid) (related to Proposition: 1.43: Complementary Segments of Parallelograms)
- Proof: 1620 BC (related to Proposition: Sum of Euler Function)
- Proof: 547 BC (related to Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Proof: 569 BC (related to Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Proof: 600 BC (related to Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Proof: 624 BC (related to Lemma: Sum of Möbius Function Over Divisors With Division)
- Proof: 800 BC (related to Proposition: Explicit Formula for the Euler Function)
- Proof: By Contradiction (related to Proposition: Partial Orders are Extensional)
- Proof: By Contradiction (related to Proposition: Strict Orders are Extensional)
- Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
- Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
- Proof: By Euclid (related to Corollary: Angles of Right Triangle)
- Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
- Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
- Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
- Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
- Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
- Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
- Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
- Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
- Proof: By Euclid (related to Corollary: Rhombus as a Special Case of a Parallelogram)
- Proof: By Euclid (related to Corollary: Similar Triangles)
- Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
- Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
- Proof: By Euclid (related to Corollary: The supplemental angle of a right angle is another right angle.)
- Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
- Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
- Proof: By Euclid (related to Proposition: 1.01: Constructing an Equilateral Triangle)
- Proof: By Euclid (related to Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment)
- Proof: By Euclid (related to Proposition: 1.03: Cutting a Segment at a Given Size)
- Proof: By Euclid (related to Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle)
- Proof: By Euclid (related to Proposition: 1.05: Isosceles Triangles I)
- Proof: By Euclid (related to Proposition: 1.06: Isosceles Triagles II)
- Proof: By Euclid (related to Proposition: 1.07: Uniqueness of Triangles)
- Proof: By Euclid (related to Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles)
- Proof: By Euclid (related to Proposition: 1.09: Bisecting an Angle)
- Proof: By Euclid (related to Proposition: 1.10: Bisecting a Segment)
- Proof: By Euclid (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Proof: By Euclid (related to Proposition: 1.14: Combining Rays to Straight Lines)
- Proof: By Euclid (related to Proposition: 1.15: Opposite Angles on Intersecting Straight Lines)
- Proof: By Euclid (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Proof: By Euclid (related to Proposition: 1.17: The Sum of Two Angles of a Triangle)
- Proof: By Euclid (related to Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality))
- Proof: By Euclid (related to Proposition: 1.21: Triangles within Triangles)
- Proof: By Euclid (related to Proposition: 1.22: Construction of Triangles From Arbitrary Segments)
- Proof: By Euclid (related to Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle)
- Proof: By Euclid (related to Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles)
- Proof: By Euclid (related to Proposition: 1.27: Parallel Lines I)
- Proof: By Euclid (related to Proposition: 1.28: Parallel Lines II)
- Proof: By Euclid (related to Proposition: 1.29: Parallel Lines III)
- Proof: By Euclid (related to Proposition: 1.30: Transitivity of Parallel Lines)
- Proof: By Euclid (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Proof: By Euclid (related to Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram)
- Proof: By Euclid (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Proof: By Euclid (related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
- Proof: By Euclid (related to Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels)
- Proof: By Euclid (related to Proposition: 1.37: Triangles of Equal Area I)
- Proof: By Euclid (related to Proposition: 1.38: Triangles of Equal Area II)
- Proof: By Euclid (related to Proposition: 1.39: Triangles of Equal Area III)
- Proof: By Euclid (related to Proposition: 1.40: Triangles of Equal Area IV)
- Proof: By Euclid (related to Proposition: 1.41: Parallelograms and Triagles)
- Proof: By Euclid (related to Proposition: 1.42: Construction of Parallelograms I)
- Proof: By Euclid (related to Proposition: 1.44: Construction of Parallelograms II)
- Proof: By Euclid (related to Proposition: 1.45: Construction of Parallelograms III)
- Proof: By Euclid (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Proof: By Euclid (related to Proposition: 1.47: Pythagorean Theorem)
- Proof: By Euclid (related to Proposition: 1.48: The Converse of the Pythagorean Theorem)
- Proof: By Euclid (related to Proposition: 2.01: Summing Areas or Rectangles)
- Proof: By Euclid (related to Proposition: 2.02: Square is Sum of Two Rectangles)
- Proof: By Euclid (related to Proposition: 2.03: Rectangle is Sum of Square and Rectangle)
- Proof: By Euclid (related to Proposition: 2.04: Square of Sum)
- Proof: By Euclid (related to Proposition: 2.05: Rectangle is Difference of Two Squares)
- Proof: By Euclid (related to Proposition: 2.06: Square of Sum with One Halved Summand)
- Proof: By Euclid (related to Proposition: 2.07: Sum of Squares)
- Proof: By Euclid (related to Proposition: 2.08: Square of Sum with One Doubled Summand)
- Proof: By Euclid (related to Proposition: 2.09: Sum of Squares of Sum and Difference)
- Proof: By Euclid (related to Proposition: 2.10: Sum of Squares (Half))
- Proof: By Euclid (related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)
- Proof: By Euclid (related to Proposition: 2.12: Law of Cosines (for Obtuse Angles))
- Proof: By Euclid (related to Proposition: 2.13: Law of Cosines (for Acute Angles))
- Proof: By Euclid (related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)
- Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
- Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Proposition: 3.01: Finding the Center of a given Circle)
- Proof: By Euclid (related to Proposition: 3.02: Chord Lies Inside its Circle)
- Proof: By Euclid (related to Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector)
- Proof: By Euclid (related to Proposition: 3.04: Chords do not Bisect Each Other)
- Proof: By Euclid (related to Proposition: 3.05: Intersecting Circles have Different Centers)
- Proof: By Euclid (related to Proposition: 3.06: Touching Circles have Different Centers)
- Proof: By Euclid (related to Proposition: 3.07: Relative Lengths of Lines Inside Circle)
- Proof: By Euclid (related to Proposition: 3.08: Relative Lengths of Lines Outside Circle)
- Proof: By Euclid (related to Proposition: 3.09: Condition for Point to be Center of Circle)
- Proof: By Euclid (related to Proposition: 3.10: Two Circles have at most Two Points of Intersection)
- Proof: By Euclid (related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
- Proof: By Euclid (related to Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally)
- Proof: By Euclid (related to Proposition: 3.13: Circles Touch at One Point at Most)
- Proof: By Euclid (related to Proposition: 3.14: Equal Chords in Circle)
- Proof: By Euclid (related to Proposition: 3.15: Relative Lengths of Chords of Circles)
- Proof: By Euclid (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Proposition: 3.17: Construction of Tangent from Point to Circle)
- Proof: By Euclid (related to Proposition: 3.18: Radius at Right Angle to Tangent)
- Proof: By Euclid (related to Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center)
- Proof: By Euclid (related to Proposition: 3.20: Inscribed Angle Theorem)
- Proof: By Euclid (related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
- Proof: By Euclid (related to Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral)
- Proof: By Euclid (related to Proposition: 3.23: Segment on Given Base Unique)
- Proof: By Euclid (related to Proposition: 3.24: Similar Segments on Equal Bases are Equal)
- Proof: By Euclid (related to Proposition: 3.25: Construction of Circle from Segment)
- Proof: By Euclid (related to Proposition: 3.26: Equal Angles and Arcs in Equal Circles)
- Proof: By Euclid (related to Proposition: 3.27: Angles on Equal Arcs are Equal)
- Proof: By Euclid (related to Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles)
- Proof: By Euclid (related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
- Proof: By Euclid (related to Proposition: 3.30: Bisection of Arc)
- Proof: By Euclid (related to Proposition: 3.31: Relative Sizes of Angles in Segments)
- Proof: By Euclid (related to Proposition: 3.32: Angles made by Chord with Tangent)
- Proof: By Euclid (related to Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle)
- Proof: By Euclid (related to Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle)
- Proof: By Euclid (related to Proposition: 3.35: Intersecting Chord Theorem)
- Proof: By Euclid (related to Proposition: 3.36: Tangent Secant Theorem)
- Proof: By Euclid (related to Proposition: 3.37: Converse of Tangent Secant Theorem)
- Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proof: By Euclid (related to Proposition: 4.01: Fitting Chord Into Circle)
- Proof: By Euclid (related to Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles)
- Proof: By Euclid (related to Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles)
- Proof: By Euclid (related to Proposition: 4.04: Inscribing Circle in Triangle)
- Proof: By Euclid (related to Proposition: 4.05: Circumscribing Circle about Triangle)
- Proof: By Euclid (related to Proposition: 4.06: Inscribing Square in Circle)
- Proof: By Euclid (related to Proposition: 4.07: Circumscribing Square about Circle)
- Proof: By Euclid (related to Proposition: 4.08: Inscribing Circle in Square)
- Proof: By Euclid (related to Proposition: 4.09: Circumscribing Circle about Square)
- Proof: By Euclid (related to Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex)
- Proof: By Euclid (related to Proposition: 4.11: Inscribing Regular Pentagon in Circle)
- Proof: By Euclid (related to Proposition: 4.12: Circumscribing Regular Pentagon about Circle)
- Proof: By Euclid (related to Proposition: 4.13: Inscribing Circle in Regular Pentagon)
- Proof: By Euclid (related to Proposition: 4.14: Circumscribing Circle about Regular Pentagon)
- Proof: By Euclid (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proof: By Euclid (related to Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle)
- Proof: By Euclid (related to Corollary: 5.07: Ratios of Equal Magnitudes)
- Proof: By Euclid (related to Corollary: 5.19: Proportional Magnitudes have Proportional Remainders)
- Proof: By Euclid (related to Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition)
- Proof: By Euclid (related to Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition)
- Proof: By Euclid (related to Proposition: 5.03: Multiplication of Numbers is Associative)
- Proof: By Euclid (related to Proposition: 5.04: Multiples of Terms in Equal Ratios)
- Proof: By Euclid (related to Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction)
- Proof: By Euclid (related to Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction)
- Proof: By Euclid (related to Proposition: 5.07: Ratios of Equal Magnitudes)
- Proof: By Euclid (related to Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes)
- Proof: By Euclid (related to Proposition: 5.09: Magnitudes with Same Ratios are Equal)
- Proof: By Euclid (related to Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios)
- Proof: By Euclid (related to Proposition: 5.11: Equality of Ratios is Transitive)
- Proof: By Euclid (related to Proposition: 5.12: Sum of Components of Equal Ratios)
- Proof: By Euclid (related to Proposition: 5.13: Relative Sizes of Proportional Magnitudes)
- Proof: By Euclid (related to Proposition: 5.14: Relative Sizes of Components of Ratios)
- Proof: By Euclid (related to Proposition: 5.15: Ratio Equals its Multiples)
- Proof: By Euclid (related to Proposition: 5.16: Proportional Magnitudes are Proportional Alternately)
- Proof: By Euclid (related to Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated)
- Proof: By Euclid (related to Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded)
- Proof: By Euclid (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
- Proof: By Euclid (related to Proposition: 5.20: Relative Sizes of Successive Ratios)
- Proof: By Euclid (related to Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion)
- Proof: By Euclid (related to Proposition: 5.22: Equality of Ratios Ex Aequali)
- Proof: By Euclid (related to Proposition: 5.23: Equality of Ratios in Perturbed Proportion)
- Proof: By Euclid (related to Proposition: 5.24: Sum of Antecedents of Proportion)
- Proof: By Euclid (related to Proposition: 5.25: Sum of Antecedent and Consequent of Proportion)
- Proof: By Euclid (related to Corollary: 6.08: Geometric Mean Theorem)
- Proof: By Euclid (related to Corollary: 6.19: Ratio of Areas of Similar Triangles)
- Proof: By Euclid (related to Corollary: 6.20: Similar Polygons are Composed of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base)
- Proof: By Euclid (related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
- Proof: By Euclid (related to Proposition: 6.03: Angle Bisector Theorem)
- Proof: By Euclid (related to Proposition: 6.04: Equiangular Triangles are Similar)
- Proof: By Euclid (related to Proposition: 6.05: Triangles with Proportional Sides are Similar)
- Proof: By Euclid (related to Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar)
- Proof: By Euclid (related to Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar)
- Proof: By Euclid (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.09: Construction of Part of Line)
- Proof: By Euclid (related to Proposition: 6.10: Construction of Similarly Cut Straight Line)
- Proof: By Euclid (related to Proposition: 6.11: Construction of Segment in Squared Ratio)
- Proof: By Euclid (related to Proposition: 6.12: Construction of Fourth Proportional Straight Line)
- Proof: By Euclid (related to Proposition: 6.13: Construction of Mean Proportional)
- Proof: By Euclid (related to Proposition: 6.14: Characterization of Congruent Parallelograms)
- Proof: By Euclid (related to Proposition: 6.15: Characterization of Congruent Triangles)
- Proof: By Euclid (related to Proposition: 6.16: Rectangles Contained by Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.18: Construction of Similar Polygon)
- Proof: By Euclid (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.21: Similarity of Polygons is Transitive)
- Proof: By Euclid (related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms)
- Proof: By Euclid (related to Proposition: 6.24: Parallelograms About Diameter are Similar)
- Proof: By Euclid (related to Proposition: 6.25: Construction of Figure Similar to One and Equal to Another)
- Proof: By Euclid (related to Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter)
- Proof: By Euclid (related to Proposition: 6.27: Similar Parallelogram on Half a Straight Line)
- Proof: By Euclid (related to Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram)
- Proof: By Euclid (related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)
- Proof: By Euclid (related to Proposition: 6.30: Construction of the Inverse Golden Section)
- Proof: By Euclid (related to Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle)
- Proof: By Euclid (related to Proposition: 6.32: Triangles with Two Sides Parallel and Equal)
- Proof: By Euclid (related to Proposition: 6.33: Angles in Circles have Same Ratio as Arcs)
- Proof: By Euclid (related to Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor)
- Proof: By Euclid (related to Proposition: 7.01: Sufficient Condition for Coprimality)
- Proof: By Euclid (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
- Proof: By Euclid (related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)
- Proof: By Euclid (related to Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers)
- Proof: By Euclid (related to Proposition: 7.05: Divisors Obey Distributive Law (Sum))
- Proof: By Euclid (related to Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum))
- Proof: By Euclid (related to Proposition: 7.07: Divisors Obey Distributive Law (Difference))
- Proof: By Euclid (related to Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference))
- Proof: By Euclid (related to Proposition: 7.09: Alternate Ratios of Equal Fractions)
- Proof: By Euclid (related to Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions)
- Proof: By Euclid (related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
- Proof: By Euclid (related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
- Proof: By Euclid (related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
- Proof: By Euclid (related to Proposition: 7.14: Proportion of Numbers is Transitive)
- Proof: By Euclid (related to Proposition: 7.15: Alternate Ratios of Multiples)
- Proof: By Euclid (related to Proposition: 7.16: Natural Number Multiplication is Commutative)
- Proof: By Euclid (related to Proposition: 7.17: Multiples of Ratios of Numbers)
- Proof: By Euclid (related to Proposition: 7.18: Ratios of Multiples of Numbers)
- Proof: By Euclid (related to Proposition: 7.19: Relation of Ratios to Products)
- Proof: By Euclid (related to Proposition: 7.20: Ratios of Fractions in Lowest Terms)
- Proof: By Euclid (related to Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms)
- Proof: By Euclid (related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime)
- Proof: By Euclid (related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
- Proof: By Euclid (related to Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole)
- Proof: By Euclid (related to Proposition: 7.25: Square of Co-prime Number is Co-prime)
- Proof: By Euclid (related to Proposition: 7.26: Product of Co-prime Pairs is Co-prime)
- Proof: By Euclid (related to Proposition: 7.27: Powers of Co-prime Numbers are Co-prime)
- Proof: By Euclid (related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)
- Proof: By Euclid (related to Proposition: 7.29: Prime not Divisor implies Co-prime)
- Proof: By Euclid (related to Proposition: 7.30: Euclidean Lemma)
- Proof: By Euclid (related to Proposition: 7.31: Existence of Prime Divisors)
- Proof: By Euclid (related to Proposition: 7.32: Natural Number is Prime or has Prime Factor)
- Proof: By Euclid (related to Proposition: 7.33: Least Ratio of Numbers)
- Proof: By Euclid (related to Proposition: 7.34: Existence of Least Common Multiple)
- Proof: By Euclid (related to Proposition: 7.35: Least Common Multiple Divides Common Multiple)
- Proof: By Euclid (related to Proposition: 7.36: Least Common Multiple of Three Numbers)
- Proof: By Euclid (related to Proposition: 7.37: Integer Divided by Divisor is Integer)
- Proof: By Euclid (related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
- Proof: By Euclid (related to Proposition: 7.39: Least Number with Three Given Fractions)
- Proof: By Euclid (related to Corollary: 8.02: Construction of Geometric Progression in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes)
- Proof: By Euclid (related to Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios)
- Proof: By Euclid (related to Proposition: 8.05: Ratio of Products of Sides of Plane Numbers)
- Proof: By Euclid (related to Proposition: 8.06: First Element of Geometric Progression not dividing Second)
- Proof: By Euclid (related to Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second)
- Proof: By Euclid (related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
- Proof: By Euclid (related to Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers)
- Proof: By Euclid (related to Proposition: Prop. 8.10: Product of Geometric Progressions from One)
- Proof: By Euclid (related to Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional)
- Proof: By Euclid (related to Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals)
- Proof: By Euclid (related to Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression)
- Proof: By Euclid (related to Proposition: Prop. 8.14: Number divides Number iff Square divides Square)
- Proof: By Euclid (related to Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
- Proof: By Euclid (related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional)
- Proof: By Euclid (related to Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals)
- Proof: By Euclid (related to Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane)
- Proof: By Euclid (related to Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid)
- Proof: By Euclid (related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)
- Proof: By Euclid (related to Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)
- Proof: By Euclid (related to Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares)
- Proof: By Euclid (related to Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes)
- Proof: By Euclid (related to Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Proof: By Euclid (related to Proposition: 9.35: Sum of Geometric Progression)
- Proof: By Euclid (related to Proposition: 9.36: Theorem of Even Perfect Numbers (First Part))
- Proof: By Euclid (related to Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square)
- Proof: By Euclid (related to Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.03: Square of Cube Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.06: Number Squared making Cube is itself Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number)
- Proof: By Euclid (related to Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Proof: By Euclid (related to Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime)
- Proof: By Euclid (related to Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime)
- Proof: By Euclid (related to Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element)
- Proof: By Euclid (related to Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional)
- Proof: By Euclid (related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second)
- Proof: By Euclid (related to Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.20: Infinite Number of Primes)
- Proof: By Euclid (related to Proposition: Prop. 9.21: Sum of Even Numbers is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.24: Even Number minus Even Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.26: Odd Number minus Odd Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.27: Odd Number minus Even Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)
- Proof: By Euclid (related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)
- Proof: By Euclid (related to Proposition: Prop. 9.32: Power of Two is Even-Times Even Only)
- Proof: By Euclid (related to Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd)
- Proof: By Euclid (related to Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic)
- Proof: By Euclid (related to Corollary: Cor. 10.009: Commensurability of Squares)
- Proof: By Euclid (related to Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial)
- Proof: By Euclid (related to Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order)
- Proof: By Euclid (related to Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas)
- Proof: By Euclid (related to Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
- Proof: By Euclid (related to Lemma: Lem. 10.021: Medial is Irrational)
- Proof: By Euclid (related to Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square)
- Proof: By Euclid (related to Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square)
- Proof: By Euclid (related to Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II)
- Proof: By Euclid (related to Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational)
- Proof: By Euclid (related to Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas)
- Proof: By Euclid (related to Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them)
- Proof: By Euclid (related to Lemma: Lem. 10.13: Finding Pythagorean Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number)
- Proof: By Euclid (related to Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm)
- Proof: By Euclid (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio)
- Proof: By Euclid (related to Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.009: Commensurability of Squares)
- Proof: By Euclid (related to Proposition: Prop. 10.010: Construction of Incommensurable Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.012: Commensurability is Transitive Relation)
- Proof: By Euclid (related to Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude)
- Proof: By Euclid (related to Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation)
- Proof: By Euclid (related to Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation)
- Proof: By Euclid (related to Proposition: Prop. 10.019: Product of Rational Numbers is Rational)
- Proof: By Euclid (related to Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational)
- Proof: By Euclid (related to Proposition: Prop. 10.021: Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.022: Square on Medial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
- Proof: By Euclid (related to Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial)
- Proof: By Euclid (related to Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square)
- Proof: By Euclid (related to Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area)
- Proof: By Euclid (related to Proposition: Prop. 10.027: Construction of Components of First Bimedial)
- Proof: By Euclid (related to Proposition: Prop. 10.028: Construction of Components of Second Bimedial)
- Proof: By Euclid (related to Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I)
- Proof: By Euclid (related to Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II)
- Proof: By Euclid (related to Proposition: Prop. 10.033: Construction of Components of Major)
- Proof: By Euclid (related to Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area)
- Proof: By Euclid (related to Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas)
- Proof: By Euclid (related to Proposition: Prop. 10.036: Binomial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.037: First Bimedial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.038: Second Bimedial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.039: Major is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.048: Construction of First Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.049: Construction of Second Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.050: Construction of Third Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order)
- Proof: By Euclid (related to Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order)
- Proof: By Euclid (related to Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major)
- Proof: By Euclid (related to Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area)
- Proof: By Euclid (related to Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas)
- Proof: By Euclid (related to Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.073: Apotome is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.074: First Apotome of Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.075: Second Apotome of Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.076: Minor is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.079: Construction of Apotome is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.082: Construction of Minor is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.085: Construction of First Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.086: Construction of Second Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.087: Construction of Third Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.088: Construction of Fourth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.089: Construction of Fifth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.090: Construction of Sixth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.103: Straight Line Commensurable with Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area)
- Proof: By Euclid (related to Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area)
- Proof: By Euclid (related to Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted)
- Proof: By Euclid (related to Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted)
- Proof: By Euclid (related to Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted)
- Proof: By Euclid (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)
- Proof: By Euclid (related to Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines)
- Proof: By Euclid (related to Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
- Proof: By Euclid (related to Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
- Proof: By Euclid (related to Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares)
- Proof: By Euclid (related to Proposition: 11.02: Two Intersecting Straight Lines are in One Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.01: Straight Line cannot be in Two Planes)
- Proof: By Euclid (related to Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other)
- Proof: By Euclid (related to Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique)
- Proof: By Euclid (related to Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes)
- Proof: By Euclid (related to Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle)
- Proof: By Euclid (related to Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle)
- Proof: By Euclid (related to Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms)
- Proof: By Euclid (related to Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes)
- Proof: By Euclid (related to Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle)
- Proof: By Euclid (related to Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped)
- Proof: By Euclid (related to Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected)
- Proof: By Euclid (related to Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume)
- Proof: By Euclid (related to Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume)
- Proof: By Euclid (related to Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume)
- Proof: By Euclid (related to Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases)
- Proof: By Euclid (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
- Proof: By Euclid (related to Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights)
- Proof: By Euclid (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme)
- Proof: By Euclid (related to Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional)
- Proof: By Euclid (related to Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube)
- Proof: By Euclid (related to Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base)
- Proof: By Euclid (related to Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
- Proof: By Euclid (related to Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Proof: By Euclid (related to Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
- Proof: By Euclid (related to Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters)
- Proof: By Euclid (related to Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
- Proof: By Euclid (related to Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters)
- Proof: By Euclid (related to Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters)
- Proof: By Euclid (related to Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms)
- Proof: By Euclid (related to Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
- Proof: By Euclid (related to Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
- Proof: By Euclid (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Proof: By Euclid (related to Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights)
- Proof: By Euclid (related to Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height)
- Proof: By Euclid (related to Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis)
- Proof: By Euclid (related to Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights)
- Proof: By Euclid (related to Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights)
- Proof: By Euclid (related to Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles)
- Proof: By Euclid (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
- Proof: By Euclid (related to Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters)
- Proof: By Euclid (related to Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere)
- Proof: By Euclid (related to Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere)
- Proof: By Euclid (related to Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere)
- Proof: By Euclid (related to Lemma: Lem. 13.18: Angle of the Pentagon)
- Proof: By Euclid (related to Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment)
- Proof: By Euclid (related to Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome)
- Proof: By Euclid (related to Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal)
- Proof: By Euclid (related to Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa)
- Proof: By Euclid (related to Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor)
- Proof: By Euclid (related to Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle)
- Proof: By Euclid (related to Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.15: Construction of Cube within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.18: There are only Five Platonic Solids)
- Proof: By Induction (related to Lemma: A Criterion for Associates)
- Proof: By Induction (related to Lemma: Continuants and Convergents)
- Proof: By Induction (related to Lemma: Fundamental Lemma of Homogeneous Systems of Linear Equations)
- Proof: By Induction (related to Proposition: Criterions for Equality of Principal Ideals)
- Proof: By Induction (related to Proposition: Generalization of Cancellative Multiplication of Integers)
- Proof: By Induction (related to Proposition: Principal Ideal Generated by A Unit)
- Proof: By Induction (related to Proposition: Principal Ideals being Maximal Ideals)
- Proof: By Induction (related to Proposition: Principal Ideals being Prime Ideals)
- Proof: By Induction (related to Proposition: Antiderivatives are Uniquely Defined Up to a Constant)
- Proof: By Induction (related to Proposition: Exponential Function of General Base With Natural Exponents)
- Proof: By Induction (related to Proposition: Generalized Bernoulli's Inequality)
- Proof: By Induction (related to Proposition: Generalized Triangle Inequality)
- Proof: By Induction (related to Proposition: Inequality between Powers of `$2$` and Factorials)
- Proof: By Induction (related to Proposition: Inequality between Square Numbers and Powers of `$2$`)
- Proof: By Induction (related to Proposition: Integrals on Adjacent Intervals)
- Proof: By Induction (related to Proposition: Limit of Nth Powers)
- Proof: By Induction (related to Proposition: Limit of a Polynomial)
- Proof: By Induction (related to Proposition: Ratio Test For Absolutely Convergent Complex Series)
- Proof: By Induction (related to Proposition: Ratio Test)
- Proof: By Induction (related to Theorem: Bernoulli's Inequality)
- Proof: By Induction (related to Theorem: De Moivre's Identity, Complex Powers)
- Proof: By Induction (related to Theorem: Every Bounded Real Sequence has a Convergent Subsequence)
- Proof: By Induction (related to Theorem: Inequality Between the Geometric and the Arithmetic Mean)
- Proof: By Induction (related to Theorem: Intermediate Root Value Theorem)
- Proof: By Induction (related to Theorem: Taylor's Formula)
- Proof: By Induction (related to Proposition: Factorial Polynomials vs. Polynomials)
- Proof: By Induction (related to Proposition: Nth Difference Operator)
- Proof: By Induction (related to Proposition: Number of Subsets of a Finite Set)
- Proof: By Induction (related to Theorem: Binomial Theorem)
- Proof: By Induction (related to Theorem: Multinomial Theorem)
- Proof: By Induction (related to Algorithm: Get the Component Induced by Vertices Connected to a Given Vertex)
- Proof: By Induction (related to Lemma: Relationship between Tree Degree, Tree Height and the Number of Leaves in a Tree)
- Proof: By Induction (related to Proposition: Chromatic Number and Maximum Vertex Degree)
- Proof: By Induction (related to Proposition: Equivalent Definitions of Trees)
- Proof: By Induction (related to Lemma: The Proving Principle by Complete Induction)
- Proof: By Induction (related to Proposition: Addition Of Natural Numbers Is Associative)
- Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Cancellative)
- Proof: By Induction (related to Proposition: Addition of Natural Numbers Is Commutative)
- Proof: By Induction (related to Proposition: Distributivity Law For Natural Numbers)
- Proof: By Induction (related to Proposition: Geometric Sum)
- Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Associative)
- Proof: By Induction (related to Proposition: Multiplication of Natural Numbers Is Cancellative)
- Proof: By Induction (related to Proposition: Multiplication of Natural Numbers is Commutative)
- Proof: By Induction (related to Proposition: Sum of Consecutive Natural Numbers)
- Proof: By Induction (related to Proposition: Sum of Consecutive Odd Numbers)
- Proof: By Induction (related to Proposition: Sum of Factorials (I))
- Proof: By Induction (related to Proposition: Existence and Number of Solutions of Congruence With One Variable)
- Proof: By Induction (related to Proposition: Existence of Solutions of an LDE With More Variables)
- Proof: By Induction (related to Proposition: Natural Numbers and Products of Prime Numbers)
- Proof: By Induction (related to Proposition: Wilson's Condition for an Integer to be Prime)
- Proof: By Induction (related to Theorem: Chinese Remainder Theorem)
- Proof: By Induction (related to Corollary: Cartesian Products of Countable Sets Is Countable)
- Proof: By Induction (related to Corollary: Circular References Of Self-Contained Sets Are Forbidden)
- Proof: By Induction (related to Proposition: Set-Theoretical Meaning of Ordered Tuples)
- Proof: Commutativity of the Greatest Common Divisor (related to Proposition: Greatest Common Divisor)
- Proof: Conformity (related to Proposition: Uniqueness of Inverse Elements)
- Proof: Uncountable Set (related to Proposition: Rational Numbers are Countable)
- Proof: What does countability mean? (related to Proposition: Real Numbers are Uncountable)
- Proof: Zero (Absorbing, Annihilating) Element, Left Zero, Right Zero (related to Proposition: Uniqueness of the Neutral Element)
- Proposition: 1.01: Constructing an Equilateral Triangle
- Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment
- Proposition: 1.03: Cutting a Segment at a Given Size
- Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
- Proposition: 1.05: Isosceles Triangles I
- Proposition: 1.06: Isosceles Triagles II
- Proposition: 1.07: Uniqueness of Triangles
- Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
- Proposition: 1.09: Bisecting an Angle
- Proposition: 1.10: Bisecting a Segment
- Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line
- Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
- Proposition: 1.13: Angles at Intersections of Straight Lines
- Proposition: 1.14: Combining Rays to Straight Lines
- Proposition: 1.15: Opposite Angles on Intersecting Straight Lines
- Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
- Proposition: 1.17: The Sum of Two Angles of a Triangle
- Proposition: 1.18: Angles and Sides in a Triangle I
- Proposition: 1.19: Angles and Sides in a Triangle II
- Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
- Proposition: 1.21: Triangles within Triangles
- Proposition: 1.22: Construction of Triangles From Arbitrary Segments
- Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
- Proposition: 1.24: Angles and Sides in a Triangle III
- Proposition: 1.25: Angles and Sides in a Triangle IV
- Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
- Proposition: 1.27: Parallel Lines I
- Proposition: 1.28: Parallel Lines II
- Proposition: 1.29: Parallel Lines III
- Proposition: 1.30: Transitivity of Parallel Lines
- Proposition: 1.31: Constructing a Parallel Line from a Line and a Point
- Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle
- Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
- Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
- Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels
- Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels
- Proposition: 1.37: Triangles of Equal Area I
- Proposition: 1.38: Triangles of Equal Area II
- Proposition: 1.39: Triangles of Equal Area III
- Proposition: 1.40: Triangles of Equal Area IV
- Proposition: 1.41: Parallelograms and Triagles
- Proposition: 1.42: Construction of Parallelograms I
- Proposition: 1.43: Complementary Segments of Parallelograms
- Proposition: 1.44: Construction of Parallelograms II
- Proposition: 1.45: Construction of Parallelograms III
- Proposition: 1.46: Construction of a Square on a Given Segment
- Proposition: 1.47: Pythagorean Theorem
- Proposition: 1.48: The Converse of the Pythagorean Theorem
- Proposition: 11.02: Two Intersecting Straight Lines are in One Plane
- Proposition: 2.01: Summing Areas or Rectangles
- Proposition: 2.02: Square is Sum of Two Rectangles
- Proposition: 2.03: Rectangle is Sum of Square and Rectangle
- Proposition: 2.04: Square of Sum
- Proposition: 2.05: Rectangle is Difference of Two Squares
- Proposition: 2.06: Square of Sum with One Halved Summand
- Proposition: 2.07: Sum of Squares
- Proposition: 2.08: Square of Sum with One Doubled Summand
- Proposition: 2.09: Sum of Squares of Sum and Difference
- Proposition: 2.10: Sum of Squares (Half)
- Proposition: 2.11: Constructing the Golden Ratio of a Segment
- Proposition: 2.12: Law of Cosines (for Obtuse Angles)
- Proposition: 2.13: Law of Cosines (for Acute Angles)
- Proposition: 2.14: Constructing a Square from a Rectilinear Figure
- Proposition: 3.01: Finding the Center of a given Circle
- Proposition: 3.02: Chord Lies Inside its Circle
- Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector
- Proposition: 3.04: Chords do not Bisect Each Other
- Proposition: 3.05: Intersecting Circles have Different Centers
- Proposition: 3.06: Touching Circles have Different Centers
- Proposition: 3.07: Relative Lengths of Lines Inside Circle
- Proposition: 3.08: Relative Lengths of Lines Outside Circle
- Proposition: 3.09: Condition for Point to be Center of Circle
- Proposition: 3.10: Two Circles have at most Two Points of Intersection
- Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally
- Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally
- Proposition: 3.13: Circles Touch at One Point at Most
- Proposition: 3.14: Equal Chords in Circle
- Proposition: 3.15: Relative Lengths of Chords of Circles
- Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle
- Proposition: 3.17: Construction of Tangent from Point to Circle
- Proposition: 3.18: Radius at Right Angle to Tangent
- Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center
- Proposition: 3.20: Inscribed Angle Theorem
- Proposition: 3.21: Angles in Same Segment of Circle are Equal
- Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral
- Proposition: 3.23: Segment on Given Base Unique
- Proposition: 3.24: Similar Segments on Equal Bases are Equal
- Proposition: 3.25: Construction of Circle from Segment
- Proposition: 3.26: Equal Angles and Arcs in Equal Circles
- Proposition: 3.27: Angles on Equal Arcs are Equal
- Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles
- Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines
- Proposition: 3.30: Bisection of Arc
- Proposition: 3.31: Relative Sizes of Angles in Segments
- Proposition: 3.32: Angles made by Chord with Tangent
- Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle
- Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle
- Proposition: 3.35: Intersecting Chord Theorem
- Proposition: 3.36: Tangent Secant Theorem
- Proposition: 3.37: Converse of Tangent Secant Theorem
- Proposition: 4.01: Fitting Chord Into Circle
- Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles
- Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles
- Proposition: 4.04: Inscribing Circle in Triangle
- Proposition: 4.05: Circumscribing Circle about Triangle
- Proposition: 4.06: Inscribing Square in Circle
- Proposition: 4.07: Circumscribing Square about Circle
- Proposition: 4.08: Inscribing Circle in Square
- Proposition: 4.09: Circumscribing Circle about Square
- Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Proposition: 4.11: Inscribing Regular Pentagon in Circle
- Proposition: 4.12: Circumscribing Regular Pentagon about Circle
- Proposition: 4.13: Inscribing Circle in Regular Pentagon
- Proposition: 4.14: Circumscribing Circle about Regular Pentagon
- Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle
- Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle
- Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition
- Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition
- Proposition: 5.03: Multiplication of Numbers is Associative
- Proposition: 5.04: Multiples of Terms in Equal Ratios
- Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction
- Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction
- Proposition: 5.07: Ratios of Equal Magnitudes
- Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes
- Proposition: 5.09: Magnitudes with Same Ratios are Equal
- Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios
- Proposition: 5.11: Equality of Ratios is Transitive
- Proposition: 5.12: Sum of Components of Equal Ratios
- Proposition: 5.13: Relative Sizes of Proportional Magnitudes
- Proposition: 5.14: Relative Sizes of Components of Ratios
- Proposition: 5.15: Ratio Equals its Multiples
- Proposition: 5.16: Proportional Magnitudes are Proportional Alternately
- Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated
- Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded
- Proposition: 5.19: Proportional Magnitudes have Proportional Remainders
- Proposition: 5.20: Relative Sizes of Successive Ratios
- Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion
- Proposition: 5.22: Equality of Ratios Ex Aequali
- Proposition: 5.23: Equality of Ratios in Perturbed Proportion
- Proposition: 5.24: Sum of Antecedents of Proportion
- Proposition: 5.25: Sum of Antecedent and Consequent of Proportion
- Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base
- Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally
- Proposition: 6.03: Angle Bisector Theorem
- Proposition: 6.04: Equiangular Triangles are Similar
- Proposition: 6.05: Triangles with Proportional Sides are Similar
- Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar
- Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
- Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles
- Proposition: 6.09: Construction of Part of Line
- Proposition: 6.10: Construction of Similarly Cut Straight Line
- Proposition: 6.11: Construction of Segment in Squared Ratio
- Proposition: 6.12: Construction of Fourth Proportional Straight Line
- Proposition: 6.13: Construction of Mean Proportional
- Proposition: 6.14: Characterization of Congruent Parallelograms
- Proposition: 6.15: Characterization of Congruent Triangles
- Proposition: 6.16: Rectangles Contained by Proportional Straight Lines
- Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines
- Proposition: 6.18: Construction of Similar Polygon
- Proposition: 6.19: Ratio of Areas of Similar Triangles
- Proposition: 6.20: Similar Polygons are Composed of Similar Triangles
- Proposition: 6.21: Similarity of Polygons is Transitive
- Proposition: 6.22: Similar Figures on Proportional Straight Lines
- Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms
- Proposition: 6.24: Parallelograms About Diameter are Similar
- Proposition: 6.25: Construction of Figure Similar to One and Equal to Another
- Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter
- Proposition: 6.27: Similar Parallelogram on Half a Straight Line
- Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
- Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
- Proposition: 6.30: Construction of the Inverse Golden Section
- Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle
- Proposition: 6.32: Triangles with Two Sides Parallel and Equal
- Proposition: 6.33: Angles in Circles have Same Ratio as Arcs
- Proposition: 7.01: Sufficient Condition for Coprimality
- Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm
- Proposition: 7.03: Greatest Common Divisor of Three Numbers
- Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers
- Proposition: 7.05: Divisors Obey Distributive Law (Sum)
- Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum)
- Proposition: 7.07: Divisors Obey Distributive Law (Difference)
- Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference)
- Proposition: 7.09: Alternate Ratios of Equal Fractions
- Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions
- Proposition: 7.11: Proportional Numbers have Proportional Differences
- Proposition: 7.12: Ratios of Numbers is Distributive over Addition
- Proposition: 7.13: Proportional Numbers are Proportional Alternately
- Proposition: 7.14: Proportion of Numbers is Transitive
- Proposition: 7.15: Alternate Ratios of Multiples
- Proposition: 7.16: Natural Number Multiplication is Commutative
- Proposition: 7.17: Multiples of Ratios of Numbers
- Proposition: 7.18: Ratios of Multiples of Numbers
- Proposition: 7.19: Relation of Ratios to Products
- Proposition: 7.20: Ratios of Fractions in Lowest Terms
- Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms
- Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime
- Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other
- Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole
- Proposition: 7.25: Square of Co-prime Number is Co-prime
- Proposition: 7.26: Product of Co-prime Pairs is Co-prime
- Proposition: 7.27: Powers of Co-prime Numbers are Co-prime
- Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both
- Proposition: 7.29: Prime not Divisor implies Co-prime
- Proposition: 7.30: Euclidean Lemma
- Proposition: 7.31: Existence of Prime Divisors
- Proposition: 7.32: Natural Number is Prime or has Prime Factor
- Proposition: 7.33: Least Ratio of Numbers
- Proposition: 7.34: Existence of Least Common Multiple
- Proposition: 7.35: Least Common Multiple Divides Common Multiple
- Proposition: 7.36: Least Common Multiple of Three Numbers
- Proposition: 7.37: Integer Divided by Divisor is Integer
- Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer
- Proposition: 7.39: Least Number with Three Given Fractions
- Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms
- Proposition: 8.02: Construction of Geometric Progression in Lowest Terms
- Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes
- Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios
- Proposition: 8.05: Ratio of Products of Sides of Plane Numbers
- Proposition: 8.06: First Element of Geometric Progression not dividing Second
- Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second
- Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements
- Proposition: 9.35: Sum of Geometric Progression
- Proposition: 9.36: Theorem of Even Perfect Numbers (First Part)
- Proposition: A Field with an Absolute Value is a Metric Space
- Proposition: A General Criterion for the Convergence of Infinite Complex Series
- Proposition: A Linear Term for 1 Using Two Co-prime Coefficients
- Proposition: A Necessary Condition for a Graph to be Planar
- Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar
- Proposition: A Necessary Condition for a Graph with Shortest Cycles to Be Planar (II)
- Proposition: A Necessary Condition for an Integer to be Prime
- Proposition: A Necessary Condition of a Neighborhood to be Open
- Proposition: A Necessary and a Sufficient Condition for Riemann Integrable Functions
- Proposition: Abel's Test
- Proposition: Abelian Group of Matrices Under Addition
- Proposition: Abelian Partial Summation Method
- Proposition: Absolute Value of Complex Conjugate
- Proposition: Absolute Value of the Product of Complex Numbers
- Proposition: Addition Of Natural Numbers
- Proposition: Addition Of Natural Numbers Is Associative
- Proposition: Addition Of Rational Numbers
- Proposition: Addition Of Real Numbers Is Associative
- Proposition: Addition Of Real Numbers Is Commutative
- Proposition: Addition of Complex Numbers Is Associative
- Proposition: Addition of Complex Numbers Is Commutative
- Proposition: Addition of Integers
- Proposition: Addition of Integers Is Associative
- Proposition: Addition of Integers Is Cancellative
- Proposition: Addition of Integers Is Commutative
- Proposition: Addition of Natural Numbers Is Cancellative
- Proposition: Addition of Natural Numbers Is Cancellative With Respect To Inequalities
- Proposition: Addition of Natural Numbers Is Commutative
- Proposition: Addition of Rational Cauchy Sequences
- Proposition: Addition of Rational Cauchy Sequences Is Associative
- Proposition: Addition of Rational Cauchy Sequences Is Cancellative
- Proposition: Addition of Rational Cauchy Sequences Is Commutative
- Proposition: Addition of Rational Numbers Is Associative
- Proposition: Addition of Rational Numbers Is Cancellative
- Proposition: Addition of Rational Numbers Is Commutative
- Proposition: Addition of Real Numbers
- Proposition: Addition of Real Numbers Is Cancellative
- Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring `$\mathbb Z_m$`
- Proposition: Additive Subgroups of Integers
- Proposition: Additivity Theorem of Tangent
- Proposition: Additivity Theorems of Cosine and Sine
- Proposition: Algebraic Structure Of Natural Numbers Together With Addition
- Proposition: Algebraic Structure Of Natural Numbers Together With Multiplication
- Proposition: Algebraic Structure of Complex Numbers Together with Addition
- Proposition: Algebraic Structure of Complex Numbers Together with Addition and Multiplication
- Proposition: Algebraic Structure of Integers Together with Addition
- Proposition: Algebraic Structure of Integers Together with Addition and Multiplication
- Proposition: Algebraic Structure of Non-Zero Complex Numbers Together with Multiplication
- Proposition: Algebraic Structure of Non-Zero Rational Numbers Together with Multiplication
- Proposition: Algebraic Structure of Non-Zero Real Numbers Together with Multiplication
- Proposition: Algebraic Structure of Rational Numbers Together with Addition
- Proposition: Algebraic Structure of Rational Numbers Together with Addition and Multiplication
- Proposition: Algebraic Structure of Real Numbers Together with Addition
- Proposition: Algebraic Structure of Real Numbers Together with Addition and Multiplication
- Proposition: All Solutions Given a Solution of an LDE With Two Variables
- Proposition: Alternating Sum of Binomial Coefficients
- Proposition: Alternative Characterization of Topological Spaces
- Proposition: Antiderivatives are Uniquely Defined Up to a Constant
- Proposition: Antidifferences are Unique Up to a Periodic Constant
- Proposition: Antidifferences of Some Functions
- Proposition: Approximation of Functions by Taylor's Formula
- Proposition: Arithmetic of Functions with Limits - Difference
- Proposition: Arithmetic of Functions with Limits - Division
- Proposition: Arithmetic of Functions with Limits - Product
- Proposition: Arithmetic of Functions with Limits - Sums
- Proposition: Associativity of Conjunction
- Proposition: Associativity of Disjunction
- Proposition: Basic Calculations Involving Indefinite Sums
- Proposition: Basic Calculations Involving the Difference Operator
- Proposition: Basic Rules of Manipulating Finite Sums
- Proposition: Basis Arithmetic Operations Involving Differentiable Functions, Product Rule, Quotient Rule
- Proposition: Bijective Open Functions
- Proposition: Binomial Distribution
- Proposition: Bounds for Partial Sums of Exponential Series
- Proposition: Calculating the Number of Positive Divisors
- Proposition: Calculating the Sum of Divisors
- Proposition: Calculating with Complex Conjugates
- Proposition: Calculation Rules for General Powers
- Proposition: Calculation Rules for the Big O Notation
- Proposition: Calculations with Uniformly Convergent Functions
- Proposition: Cancellation Law
- Proposition: Cancellation of Congruences With Factor Co-Prime To Module, Field `$\mathbb Z_p$`
- Proposition: Cancellation of Congruences with General Factor
- Proposition: Cardinal Number
- Proposition: Cardinals of a Set and Its Power Set
- Proposition: Cauchy Condensation Criterion
- Proposition: Cauchy Criterion
- Proposition: Cauchy Product of Absolutely Convergent Complex Series
- Proposition: Cauchy Product of Absolutely Convergent Series
- Proposition: Cauchy Product of Convergent Series Is Not Necessarily Convergent
- Proposition: Cauchy-Schwarz Inequality for Integral p-norms
- Proposition: Cauchy-Schwarz Test
- Proposition: Cauchy–Schwarz Inequality
- Proposition: Chain Rule
- Proposition: Characterization of Bijective Functions
- Proposition: Characterization of Cutvertices
- Proposition: Characterization of Dependent Absolute Values
- Proposition: Characterization of Independent Events
- Proposition: Characterization of Independent Events II
- Proposition: Characterization of Monotonic Functions via Derivatives
- Proposition: Characterization of Non-Archimedean Absolute Values
- Proposition: Characterization of `$T_1$` Spaces
- Proposition: Characterization of `$T_2$` Spaces
- Proposition: Chromatic Number and Maximum Vertex Degree
- Proposition: Clopen Sets and Boundaries
- Proposition: Closed Formula For Binomial Coefficients
- Proposition: Closed Formula for the Maximum and Minimum of Two Numbers
- Proposition: Closed Subsets of Compact Sets are Compact
- Proposition: Closed n-Dimensional Cuboids Are Compact
- Proposition: Co-prime Primes
- Proposition: Common Points of Two Distinct Straight Lines in a Plane
- Proposition: Common Points of a Plane and a Straight Line Not in the Plane
- Proposition: Compact Subset of Real Numbers Contains its Maximum and its Minimum
- Proposition: Compact Subsets of Metric Spaces Are Bounded and Closed
- Proposition: Comparing Natural Numbers Using the Concept of Addition
- Proposition: Comparison between the Stirling numbers of the First and Second Kind
- Proposition: Comparison of Functional Equations For Linear, Logarithmic and Exponential Growth
- Proposition: Complete and Reduced Residue Systems (Revised)
- Proposition: Complex Cauchy Sequences Vs. Real Cauchy Sequences
- Proposition: Complex Conjugate of Complex Exponential Function
- Proposition: Complex Convergent Sequences are Bounded
- Proposition: Complex Exponential Function
- Proposition: Complex Numbers Cannot Be Ordered
- Proposition: Complex Numbers are a Field Extension of Real Numbers
- Proposition: Complex Numbers as a Vector Space Over the Field of Real Numbers
- Proposition: Composition of Bijective Functions is Bijective
- Proposition: Composition of Continuous Functions at a Single Point
- Proposition: Composition of Functions is Associative
- Proposition: Composition of Injective Functions is Injective
- Proposition: Composition of Surjective Functions is Surjective
- Proposition: Compositions of Continuous Functions on a Whole Domain
- Proposition: Congruence Classes
- Proposition: Congruence Modulo a Divisor
- Proposition: Congruences and Division with Quotient and Remainder
- Proposition: Connection between Quotient, Remainder, Modulo and Floor Function
- Proposition: Connectivity Is an Equivalence Relation - Components Are a Partition of a Graph
- Proposition: Construction of Topological Spaces Using a Subbasis
- Proposition: Construction of a Light Clock
- Proposition: Contained Relation is a Strict Order
- Proposition: Continuity of Complex Exponential Function
- Proposition: Continuity of Compositions of Functions
- Proposition: Continuity of Cosine and Sine
- Proposition: Continuity of Exponential Function
- Proposition: Continuity of Exponential Function of General Base
- Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals
- Proposition: Continuous Real Functions on Closed Intervals are Riemann-Integrable
- Proposition: Contraposition of Cancellative Law for Adding Integers
- Proposition: Contraposition of Cancellative Law for Adding Rational Numbers
- Proposition: Contraposition of Cancellative Law for Adding Real Numbers
- Proposition: Contraposition of Cancellative Law for Multiplying Integers
- Proposition: Contraposition of Cancellative Law for Multiplying Natural Numbers
- Proposition: Contraposition of Cancellative Law for Multiplying Rational Numbers
- Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers
- Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Infinity
- Proposition: Convergence Behavior of the Inverse of Sequence Members Tending to Zero
- Proposition: Convergence Behavior of the Sequence `\((b^n)\)`
- Proposition: Convergence Behaviour of Absolutely Convergent Series
- Proposition: Convergence of Alternating Harmonic Series
- Proposition: Convergence of Series Implies Sequence of Terms Converges to Zero
- Proposition: Convergent Complex Sequences Are Bounded
- Proposition: Convergent Complex Sequences Are Cauchy Sequences
- Proposition: Convergent Complex Sequences Vs. Convergent Real Sequences
- Proposition: Convergent Real Sequences Are Cauchy Sequences
- Proposition: Convergent Real Sequences are Bounded
- Proposition: Convergent Sequence together with Limit is a Compact Subset of Metric Space
- Proposition: Convergent Sequence without Limit Is Not a Compact Subset of Metric Space
- Proposition: Convergent Sequences are Bounded
- Proposition: Convex Functions on Open Intervals are Continuous
- Proposition: Convexity and Concaveness Test
- Proposition: Counting the Roots of a Diophantine Polynomial Modulo a Prime Number
- Proposition: Counting the Set's Elements Using Its Partition
- Proposition: Counting the Solutions of Diophantine Equations of Congruences
- Proposition: Creation of Complete Residue Systems From Others
- Proposition: Creation of Reduced Residue Systems From Others
- Proposition: Criteria for Subgroups
- Proposition: Criterions for Equality of Principal Ideals
- Proposition: De Morgan's Laws (Sets)
- Proposition: Definition of Integers
- Proposition: Definition of Rational Numbers
- Proposition: Definition of Real Numbers
- Proposition: Definition of the Metric Space `\(\mathbb R^n\)`, Euclidean Norm
- Proposition: Derivate of Absolute Value Function Does Not Exist at `\(0\)`
- Proposition: Derivative of Cosine
- Proposition: Derivative of General Powers of Positive Numbers
- Proposition: Derivative of Sine
- Proposition: Derivative of Tangent
- Proposition: Derivative of an Invertible Function on Real Invervals
- Proposition: Derivative of the Exponential Function
- Proposition: Derivative of the Inverse Sine
- Proposition: Derivative of the Inverse Tangent
- Proposition: Derivative of the Natural Logarithm
- Proposition: Derivative of the Reciprocal Function
- Proposition: Derivative of the n-th Power Function
- Proposition: Derivatives of Even and Odd Functions
- Proposition: Difference Operator of Falling Factorial Powers
- Proposition: Difference Operator of Powers
- Proposition: Difference of Convergent Complex Sequences
- Proposition: Difference of Convergent Real Sequences
- Proposition: Difference of Convergent Real Series
- Proposition: Difference of Squares of Hyperbolic Cosine and Hyperbolic Sine
- Proposition: Differentiable Functions and Tangent-Linear Approximation
- Proposition: Differential Equation of the Exponential Function
- Proposition: Diophantine Equations of Congruences
- Proposition: Direct Comparison Test For Absolutely Convergent Complex Series
- Proposition: Direct Comparison Test For Absolutely Convergent Series
- Proposition: Direct Comparison Test For Divergent Series
- Proposition: Dirichlet's Test
- Proposition: Discovery of Irrational Numbers
- Proposition: Distance in Normed Vector Spaces
- Proposition: Distributivity Law For Integers
- Proposition: Distributivity Law For Natural Numbers
- Proposition: Distributivity Law For Rational Cauchy Sequences
- Proposition: Distributivity Law For Rational Numbers
- Proposition: Distributivity Law For Real Numbers
- Proposition: Distributivity Law for Complex Numbers
- Proposition: Distributivity Laws For Sets
- Proposition: Divergence of Harmonic Series
- Proposition: Divisibility Laws
- Proposition: Divisors of a Product Of Two Factors, Co-Prime to One Factor Divide the Other Factor
- Proposition: Double Summation
- Proposition: Equality of Two Ratios
- Proposition: Equivalent Definitions of Trees
- Proposition: Equivalent Knot Diagrams
- Proposition: Equivalent Notions of Continuous Functions
- Proposition: Equivalent Notions of Homeomorphisms
- Proposition: Equivalent Notions of Ordinals
- Proposition: Estimate for the Remainder Term of Complex Exponential Function
- Proposition: Estimate for the Remainder Term of Exponential Function
- Proposition: Estimates for the Remainder Terms of the Infinite Series of Cosine and Sine
- Proposition: Euler's Criterion For Quadratic Residues
- Proposition: Euler's Formula
- Proposition: Even Perfect Numbers
- Proposition: Eveness (Oddness) of Polynomials
- Proposition: Eveness of the Cosine of a Real Variable
- Proposition: Every Integer Is Either Even or Odd
- Proposition: Every Natural Number Is Greater or Equal Zero
- Proposition: Existence and Number of Solutions of Congruence With One Variable
- Proposition: Existence and Uniqueness of Greatest Elements in Subsets of Natural Numbers
- Proposition: Existence of Complex One (Neutral Element of Multiplication of Complex Numbers)
- Proposition: Existence of Complex Zero (Neutral Element of Addition of Complex Numbers)
- Proposition: Existence of Integer One (Neutral Element of Multiplication of Integers)
- Proposition: Existence of Integer Zero (Neutral Element of Addition of Integers)
- Proposition: Existence of Inverse Complex Numbers With Respect to Addition
- Proposition: Existence of Inverse Complex Numbers With Respect to Multiplication
- Proposition: Existence of Inverse Integers With Respect to Addition
- Proposition: Existence of Inverse Rational Cauchy Sequences With Respect to Addition
- Proposition: Existence of Inverse Rational Numbers With Respect to Addition
- Proposition: Existence of Inverse Rational Numbers With Respect to Multiplication
- Proposition: Existence of Inverse Real Numbers With Respect to Addition
- Proposition: Existence of Inverse Real Numbers With Respect to Multiplication
- Proposition: Existence of Prime Divisors
- Proposition: Existence of Rational Cauchy Sequence of Ones (Neutral Element of Multiplication of Rational Cauchy Sequences)
- Proposition: Existence of Rational Cauchy Sequence of Zeros (Neutral Element of Addition of Rational Cauchy Sequences)
- Proposition: Existence of Rational One (Neutral Element of Multiplication of Rational Numbers)
- Proposition: Existence of Rational Zero (Neutral Element of Addition of Rational Numbers)
- Proposition: Existence of Real One (Neutral Element of Multiplication of Real Numbers)
- Proposition: Existence of Real Zero (Neutral Element of Addition of Real Numbers)
- Proposition: Existence of Solutions of an LDE With More Variables
- Proposition: Explicit Formula for the Euler Function
- Proposition: Exponential Function
- Proposition: Exponential Function of General Base With Integer Exponents
- Proposition: Exponential Function of General Base With Natural Exponents
- Proposition: Extracting the Real and the Imaginary Part of a Complex Number
- Proposition: Factorial
- Proposition: Factorial Polynomials have a Unique Representation
- Proposition: Factorial Polynomials vs. Polynomials
- Proposition: Factorials and Stirling Numbers of the First Kind
- Proposition: Factorization of Greatest Common Divisor and Least Common Multiple
- Proposition: Filter Base
- Proposition: Finite Chains are Well-ordered
- Proposition: Finite Number of Divisors
- Proposition: Finite Order of an Element Equals Order Of Generated Group
- Proposition: Fixed-Point Property of Continuous Functions on Closed Intervals
- Proposition: Floor Function and Division with Quotient and Remainder
- Proposition: Functional Equation of the Complex Exponential Function
- Proposition: Functional Equation of the Exponential Function
- Proposition: Functional Equation of the Exponential Function of General Base
- Proposition: Functional Equation of the Exponential Function of General Base (Revised)
- Proposition: Functional Equation of the Natural Logarithm
- Proposition: Functions Constitute Equivalence Relations
- Proposition: Fundamental Counting Principle
- Proposition: Gamma Function
- Proposition: Gamma Function Interpolates the Factorial
- Proposition: General Powers of Positive Numbers
- Proposition: Generalization of Cancellative Multiplication of Integers
- Proposition: Generalized Bernoulli's Inequality
- Proposition: Generalized Product Rule
- Proposition: Generalized Triangle Inequality
- Proposition: Generating Co-Prime Numbers Knowing the Greatest Common Divisor
- Proposition: Generating the Greatest Common Divisor Knowing Co-Prime Numbers
- Proposition: Geometric Distribution
- Proposition: Geometric Sum
- Proposition: Greatest Common Divisor
- Proposition: Greatest Common Divisor of More Than Two Numbers
- Proposition: Greatest Common Divisors Of Integers and Prime Numbers
- Proposition: Group Homomorphisms with Cyclic Groups
- Proposition: Group of Units
- Proposition: How Convergence Preserves Upper and Lower Bounds For Sequence Members
- Proposition: How Convergence Preserves the Order Relation of Sequence Members
- Proposition: How the Boundary Changes the Property of a Set of Being Open
- Proposition: Hölder's Inequality
- Proposition: Hölder's Inequality for Integral p-norms
- Proposition: Identity Function is Continuous
- Proposition: Image of a Compact Set Under a Continuous Function
- Proposition: Imaginary Unit
- Proposition: In a Field, `$0$` Is Unequal `$1$`
- Proposition: Indicator Function and Set Operations
- Proposition: Inequality between Binomial Coefficients and Reciprocals of Factorials
- Proposition: Inequality between Powers of `$2$` and Factorials
- Proposition: Inequality between Square Numbers and Powers of `$2$`
- Proposition: Inequality of Natural Numbers and Their Successors
- Proposition: Infinite Geometric Series
- Proposition: Infinite Series for Cosine and Sine
- Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function
- Proposition: Infinitesimal Growth of Sine is the Growth of the Identity Function
- Proposition: Inheritance of the `$T_1$` Property
- Proposition: Inheritance of the `$T_2$` Property
- Proposition: Injective, Surjective and Bijective Compositions
- Proposition: Integral Test for Convergence
- Proposition: Integral of Cosine
- Proposition: Integral of General Powers
- Proposition: Integral of Inverse Sine
- Proposition: Integral of Sine
- Proposition: Integral of the Exponential Function
- Proposition: Integral of the Inverse Tangent
- Proposition: Integral of the Natural Logarithm
- Proposition: Integral of the Reciprocal Function
- Proposition: Integral p-Norm
- Proposition: Integrals on Adjacent Intervals
- Proposition: Intersection of a Set With Another Set is Subset of This Set
- Proposition: Inverse Cosine of a Real Variable
- Proposition: Inverse Hyperbolic Cosine
- Proposition: Inverse Hyperbolic Sine
- Proposition: Inverse Sine of a Real Variable
- Proposition: Inverse Tangent and Complex Exponential Function
- Proposition: Inverse Tangent of a Real Variable
- Proposition: Inversion Formulas For Stirling Numbers
- Proposition: Isometry is Injective
- Proposition: Least Common Multiple
- Proposition: Least Common Multiple of More Than Two Numbers
- Proposition: Legendre Polynomials and Legendre Differential Equations
- Proposition: Legendre Symbols of Equal Residues
- Proposition: Leibniz Criterion for Alternating Series
- Proposition: Limit Comparizon Test
- Proposition: Limit Inferior is the Infimum of Accumulation Points of a Bounded Real Sequence
- Proposition: Limit Superior is the Supremum of Accumulation Points of a Bounded Real Sequence
- Proposition: Limit Test for Roots or Ratios
- Proposition: Limit of 1/n
- Proposition: Limit of Exponential Growth as Compared to Polynomial Growth
- Proposition: Limit of Logarithmic Growth as Compared to Positive Power Growth
- Proposition: Limit of Nested Real Intervals
- Proposition: Limit of Nth Powers
- Proposition: Limit of Nth Root of N
- Proposition: Limit of Nth Root of a Positive Constant
- Proposition: Limit of a Function is Unique If It Exists
- Proposition: Limit of a Polynomial
- Proposition: Limit of a Rational Function
- Proposition: Limit of the Constant Function
- Proposition: Limit of the Identity Function
- Proposition: Limits of General Powers
- Proposition: Limits of Logarithm in `$[0,+\infty]$`
- Proposition: Limits of Polynomials at Infinity
- Proposition: Linearity and Monotony of the Riemann Integral
- Proposition: Linearity and Monotony of the Riemann Integral for Step Functions
- Proposition: Logarithm to a General Base
- Proposition: Maximum Norm as a Limit of p-Norms
- Proposition: Metric Spaces and Empty Sets are Clopen
- Proposition: Metric Spaces are Hausdorff Spaces
- Proposition: Minkowski's Inequality
- Proposition: Minkowski's Inequality for Integral p-norms
- Proposition: Modulus of Continuity is Continuous
- Proposition: Modulus of Continuity is Monotonically Increasing
- Proposition: Modulus of Continuity is Subadditive
- Proposition: Monotonic Real Functions on Closed Intervals are Riemann-Integrable
- Proposition: Monotonically Increasing Property of Probability Distributions
- Proposition: Monotony Criterion
- Proposition: More Characterizations of Finite Sets
- Proposition: Multinomial Coefficient
- Proposition: Multinomial Distribution
- Proposition: Multiplication Of Rational Cauchy Sequences
- Proposition: Multiplication Of Rational Numbers
- Proposition: Multiplication Of Rational Numbers Is Cancellative
- Proposition: Multiplication Of Rational Numbers Is Commutative
- Proposition: Multiplication of Complex Numbers Is Associative
- Proposition: Multiplication of Complex Numbers Is Commutative
- Proposition: Multiplication of Complex Numbers Using Polar Coordinates
- Proposition: Multiplication of Congruences with a Positive Factor
- Proposition: Multiplication of Integers
- Proposition: Multiplication of Integers Is Associative
- Proposition: Multiplication of Integers Is Cancellative
- Proposition: Multiplication of Integers Is Commutative
- Proposition: Multiplication of Natural Numbers Is Associative
- Proposition: Multiplication of Natural Numbers Is Cancellative
- Proposition: Multiplication of Natural Numbers Is Cancellative With Respect to the Order Relation
- Proposition: Multiplication of Natural Numbers is Commutative
- Proposition: Multiplication of Rational Cauchy Sequences Is Associative
- Proposition: Multiplication of Rational Cauchy Sequences Is Cancellative
- Proposition: Multiplication of Rational Cauchy Sequences Is Commutative
- Proposition: Multiplication of Rational Numbers Is Associative
- Proposition: Multiplication of Real Numbers
- Proposition: Multiplication of Real Numbers Is Associative
- Proposition: Multiplication of Real Numbers Is Cancellative
- Proposition: Multiplication of Real Numbers Is Commutative
- Proposition: Multiplicative Group Modulo an Integer `$(\mathbb Z_m^*,\cdot)$`
- Proposition: Multiplicativity of the Legendre Symbol
- Proposition: Multiplying Negative and Positive Integers
- Proposition: Multiplying Negative and Positive Rational Numbers
- Proposition: Multiplying Negative and Positive Real Numbers
- Proposition: Natural Logarithm
- Proposition: Natural Logarithm Sum of von Mangoldt Function Over Divisors
- Proposition: Natural Numbers and Products of Prime Numbers
- Proposition: Not all Cauchy Sequences Converge in the set of Rational Numbers
- Proposition: Not all Continuous Functions are also Uniformly Continuous
- Proposition: Nth Difference Operator
- Proposition: Nth Powers
- Proposition: Nth Roots of Positive Numbers
- Proposition: Number of Multiples of a Given Number Less Than Another Number
- Proposition: Number of Ordered n-Tuples in a Set
- Proposition: Number of Quadratic Residues in Reduced Residue Systems Modulo a Prime
- Proposition: Number of Relations on a Finite Set
- Proposition: Number of Strings With a Fixed Length Over an Alphabet with k Letters
- Proposition: Number of Subsets of a Finite Set
- Proposition: Numbers Being the Product of Their Divisors
- Proposition: Oddness of the Sine of a Real Variable
- Proposition: Only the Uniform Convergence Preserves Continuity
- Proposition: Open Intervals Contain Uncountably Many Irrational Numbers
- Proposition: Open Real Intervals are Uncountable
- Proposition: Open and Closed Subsets of a Zariski Topology
- Proposition: Order Relation for Integers is Strict Total
- Proposition: Order Relation for Natural Numbers, Revised
- Proposition: Order Relation for Rational Numbers is Strict Total
- Proposition: Order Relation for Real Numbers is Strict and Total
- Proposition: Ordinals Are Downward Closed
- Proposition: Partial Orders are Extensional
- Proposition: Perfect Sets vs. Derived Sets
- Proposition: Plane Determined by a Straight Line and a Point not on the Straight Line
- Proposition: Plane Determined by two Crossing Straight Lines
- Proposition: Polar Coordinates of a Complex Number
- Proposition: Position of Minus Sign in Rational Numbers Representations
- Proposition: Positive and Negative Parts of a Riemann-Integrable Functions are Riemann-Integrable
- Proposition: Presentation of a Straight Line in a Plane as a Linear Equation
- Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions
- Proposition: Preservation of Continuity with Arithmetic Operations on Continuous Functions on a Whole Domain
- Proposition: Preservation of Inequalities for Limits of Functions
- Proposition: Principal Ideal Generated by A Unit
- Proposition: Principal Ideals being Maximal Ideals
- Proposition: Principal Ideals being Prime Ideals
- Proposition: Probability of Event Difference
- Proposition: Probability of Event Union
- Proposition: Probability of Included Event
- Proposition: Probability of Joint Events
- Proposition: Probability of the Complement Event
- Proposition: Product of Convegent Complex Sequences
- Proposition: Product of Convegent Real Sequences
- Proposition: Product of Riemann-integrable Functions is Riemann-integrable
- Proposition: Product of Two Even Numbers
- Proposition: Product of Two Odd Numbers
- Proposition: Product of Two Ratios
- Proposition: Product of Two Sums (Generalized Distributivity Rule)
- Proposition: Product of a Complex Number and a Convergent Complex Sequence
- Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity
- Proposition: Product of a Real Number and a Convergent Real Sequence
- Proposition: Product of a Real Number and a Convergent Real Series
- Proposition: Product of an Even and an Odd Number
- Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number
- Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm
- Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes
- Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes
- Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes
- Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable
- Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio
- Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable
- Proposition: Prop. 10.009: Commensurability of Squares
- Proposition: Prop. 10.010: Construction of Incommensurable Lines
- Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes
- Proposition: Prop. 10.012: Commensurability is Transitive Relation
- Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
- Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines
- Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes
- Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation
- Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation
- Proposition: Prop. 10.019: Product of Rational Numbers is Rational
- Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational
- Proposition: Prop. 10.021: Medial is Irrational
- Proposition: Prop. 10.022: Square on Medial Straight Line
- Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial
- Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
- Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
- Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area
- Proposition: Prop. 10.027: Construction of Components of First Bimedial
- Proposition: Prop. 10.028: Construction of Components of Second Bimedial
- Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable
- Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable
- Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I
- Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II
- Proposition: Prop. 10.033: Construction of Components of Major
- Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area
- Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas
- Proposition: Prop. 10.036: Binomial is Irrational
- Proposition: Prop. 10.037: First Bimedial is Irrational
- Proposition: Prop. 10.038: Second Bimedial is Irrational
- Proposition: Prop. 10.039: Major is Irrational
- Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational
- Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational
- Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely
- Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely
- Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely
- Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely
- Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely
- Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely
- Proposition: Prop. 10.048: Construction of First Binomial Straight Line
- Proposition: Prop. 10.049: Construction of Second Binomial Straight Line
- Proposition: Prop. 10.050: Construction of Third Binomial Straight Line
- Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line
- Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line
- Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line
- Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial
- Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial
- Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial
- Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial
- Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial
- Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial
- Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
- Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
- Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major
- Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area
- Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas
- Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
- Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
- Proposition: Prop. 10.073: Apotome is Irrational
- Proposition: Prop. 10.074: First Apotome of Medial is Irrational
- Proposition: Prop. 10.075: Second Apotome of Medial is Irrational
- Proposition: Prop. 10.076: Minor is Irrational
- Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational
- Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational
- Proposition: Prop. 10.079: Construction of Apotome is Unique
- Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique
- Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique
- Proposition: Prop. 10.082: Construction of Minor is Unique
- Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique
- Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique
- Proposition: Prop. 10.085: Construction of First Apotome
- Proposition: Prop. 10.086: Construction of Second Apotome
- Proposition: Prop. 10.087: Construction of Third Apotome
- Proposition: Prop. 10.088: Construction of Fourth Apotome
- Proposition: Prop. 10.089: Construction of Fifth Apotome
- Proposition: Prop. 10.090: Construction of Sixth Apotome
- Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome
- Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome
- Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome
- Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome
- Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome
- Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome
- Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line
- Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
- Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.103: Straight Line Commensurable with Apotome
- Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line
- Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line
- Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area
- Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area
- Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted
- Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
- Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
- Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line
- Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line
- Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome
- Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
- Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines
- Proposition: Prop. 11.01: Straight Line cannot be in Two Planes
- Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line
- Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
- Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane
- Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel
- Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane
- Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
- Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other
- Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
- Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane
- Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane
- Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique
- Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel
- Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel
- Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel
- Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes
- Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
- Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
- Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
- Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles
- Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
- Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles
- Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
- Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes
- Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle
- Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped
- Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
- Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
- Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
- Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
- Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases
- Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
- Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
- Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
- Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme
- Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
- Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base
- Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
- Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters
- Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases
- Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
- Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra
- Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides
- Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height
- Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
- Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
- Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
- Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
- Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
- Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres
- Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters
- Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
- Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
- Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal
- Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
- Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
- Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa
- Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
- Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
- Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere
- Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere
- Proposition: Prop. 13.15: Construction of Cube within Given Sphere
- Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere
- Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere
- Proposition: Prop. 13.18: There are only Five Platonic Solids
- Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers
- Proposition: Prop. 8.10: Product of Geometric Progressions from One
- Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional
- Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals
- Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression
- Proposition: Prop. 8.14: Number divides Number iff Square divides Square
- Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube
- Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square
- Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube
- Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional
- Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals
- Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane
- Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid
- Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square
- Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube
- Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square
- Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
- Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares
- Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes
- Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square
- Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers
- Proposition: Prop. 9.03: Square of Cube Number is Cube
- Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube
- Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube
- Proposition: Prop. 9.06: Number Squared making Cube is itself Cube
- Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number
- Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number
- Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number
- Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number
- Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements
- Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime
- Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime
- Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element
- Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional
- Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second
- Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers
- Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers
- Proposition: Prop. 9.20: Infinite Number of Primes
- Proposition: Prop. 9.21: Sum of Even Numbers is Even
- Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even
- Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd
- Proposition: Prop. 9.24: Even Number minus Even Number is Even
- Proposition: Prop. 9.25: Even Number minus Odd Number is Odd
- Proposition: Prop. 9.26: Odd Number minus Odd Number is Even
- Proposition: Prop. 9.27: Odd Number minus Even Number is Odd
- Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even
- Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd
- Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half
- Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double
- Proposition: Prop. 9.32: Power of Two is Even-Times Even Only
- Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd
- Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
- Proposition: Properties of Cosets
- Proposition: Properties of Floors and Ceilings
- Proposition: Properties of a Complex Scalar Product
- Proposition: Properties of a Group Homomorphism
- Proposition: Properties of the Set of All Neighborhoods of a Point
- Proposition: Pythagorean Identity
- Proposition: Quadratic Formula
- Proposition: Quotient Space
- Proposition: Quotient of Convergent Complex Sequences
- Proposition: Quotient of Convergent Real Sequences
- Proposition: Raabe's Test
- Proposition: Ratio Test
- Proposition: Ratio Test For Absolutely Convergent Complex Series
- Proposition: Ratio of Two Ratios
- Proposition: Rational Cauchy Sequence Members Are Bounded
- Proposition: Rational Functions are Continuous
- Proposition: Rational Numbers are Countable
- Proposition: Rational Numbers are Dense in Real Numbers
- Proposition: Rational Powers of Positive Numbers
- Proposition: Real Numbers are Uncountable
- Proposition: Real Sequences Contain Monotonic Subsequences
- Proposition: Rearrangement of Absolutely Convergent Series
- Proposition: Rearrangement of Convergent Series
- Proposition: Recursive Formula for Binomial Coefficients
- Proposition: Recursive Formula for the Stirling Numbers of the First Kind
- Proposition: Recursive Formula for the Stirling Numbers of the Second Kind
- Proposition: Recursively Defined Arithmetic Functions, Recursion
- Proposition: Relationship Between Planarity and Biconnectivity of Graphs
- Proposition: Relationship Between Planarity and Connectivity of Graphs
- Proposition: Relationship Between the Greatest Common Divisor and the Least Common Multiple
- Proposition: Relationship between Limit, Limit Superior, and Limit Inferior of a Real Sequence
- Proposition: Replacing Mutually Independent Events by Their Complements
- Proposition: Riemann Integral for Step Functions
- Proposition: Riemann Sum Converging To the Riemann Integral
- Proposition: Riemann Upper and Riemann Lower Integrals for Bounded Real Functions
- Proposition: Root Test
- Proposition: Rule of Combining Different Sets of Indices
- Proposition: Set Intersection is Associative
- Proposition: Set Intersection is Commutative
- Proposition: Set Union is Associative
- Proposition: Set Union is Commutative
- Proposition: Set-Theoretical Meaning of Ordered Tuples
- Proposition: Sets and Their Complements
- Proposition: Sets are Subsets of Their Union
- Proposition: Sign of Divisors of Integers
- Proposition: Simple Binomial Identities
- Proposition: Simple Calculations Rules in a Group
- Proposition: Simple Consequences from the Definition of a Vector Space
- Proposition: Special Values for Real Sine, Real Cosine and Complex Exponential Function
- Proposition: Spectrum Function of Commutative Rings
- Proposition: Square Roots
- Proposition: Square of a Non-Zero Element is Positive in Ordered Fields
- Proposition: Step Function on Closed Intervals are Riemann-Integrable
- Proposition: Step Functions as a Subspace of all Functions on a Closed Real Interval
- Proposition: Strict Orders are Extensional
- Proposition: Subgroups of Finite Cyclic Groups
- Proposition: Subset of Powers is a Submonoid
- Proposition: Subset of a Countable Set is Countable
- Proposition: Subsets of Finite Sets
- Proposition: Sufficient Condition for a Local Extremum
- Proposition: Sum and Difference of Two Ratios
- Proposition: Sum of Arguments of Hyperbolic Cosine
- Proposition: Sum of Arguments of Hyperbolic Sine
- Proposition: Sum of Arithmetic Progression
- Proposition: Sum of Binomial Coefficients
- Proposition: Sum of Binomial Coefficients I
- Proposition: Sum of Binomial Coefficients II
- Proposition: Sum of Binomial Coefficients III
- Proposition: Sum of Binomial Coefficients IV
- Proposition: Sum of Consecutive Natural Numbers
- Proposition: Sum of Consecutive Odd Numbers
- Proposition: Sum of Convergent Complex Sequences
- Proposition: Sum of Convergent Real Sequences
- Proposition: Sum of Convergent Real Series
- Proposition: Sum of Cosines
- Proposition: Sum of Cube Numbers
- Proposition: Sum of Euler Function
- Proposition: Sum of Factorials (I)
- Proposition: Sum of Geometric Progression
- Proposition: Sum of Möbius Function Over Divisors
- Proposition: Sum of Squares
- Proposition: Sum of a Convergent Real Sequence and a Real Sequence Tending to Infininty
- Proposition: Supremum Norm and Uniform Convergence
- Proposition: Taylor's Formula with Remainder Term of Lagrange
- Proposition: The Contained Relation is Extensional
- Proposition: The Equality of Sets Is an Equivalence Relation
- Proposition: The General Perturbation Method
- Proposition: The Inverse Of a Composition
- Proposition: The distance of complex numbers makes complex numbers a metric space.
- Proposition: The distance of real numbers makes real numbers a metric space.
- Proposition: Time Dilation, Lorentz Factor
- Proposition: Transitive Recursion
- Proposition: Transitivity of the Order Relation of Natural Numbers
- Proposition: Uncountable and Countable Subsets of Natural Numbers
- Proposition: Uniform Convergence Criterion of Cauchy
- Proposition: Uniform Convergence Criterion of Weierstrass for Infinite Series
- Proposition: Union of Countably Many Countable Sets
- Proposition: Unique Representation of Real Numbers as `\(b\)`-adic Fractions
- Proposition: Unique Solvability of `$a\ast x=b$` in Groups
- Proposition: Unique Solvability of `$ax=b$`
- Proposition: Unique Solvability of `\(a+x=b\)`
- Proposition: Uniqueness Of Natural One
- Proposition: Uniqueness Of Predecessors Of Natural Numbers
- Proposition: Uniqueness Of Rational One
- Proposition: Uniqueness Of the Limit of a Sequence
- Proposition: Uniqueness of Complex Zero
- Proposition: Uniqueness of Integer One
- Proposition: Uniqueness of Integer Zero
- Proposition: Uniqueness of Inverse Elements
- Proposition: Uniqueness of Inverse Rational Numbers With Respect to Multiplication
- Proposition: Uniqueness of Inverse Real Numbers With Respect to Multiplication
- Proposition: Uniqueness of Natural Zero
- Proposition: Uniqueness of Negative Numbers
- Proposition: Uniqueness of Rational Zero
- Proposition: Uniqueness of Real One
- Proposition: Uniqueness of Real Zero
- Proposition: Uniqueness of the Limit of a Sequence
- Proposition: Uniqueness of the Neutral Element
- Proposition: Urn Model With Replacement
- Proposition: Urn Model Without Replacement
- Proposition: Well-Ordering Principle of Natural Numbers
- Proposition: Well-ordered Sets are Chains
- Proposition: Wilson's Condition for an Integer to be Prime
- Proposition: Zero of Cosine
- Proposition: Zero-Derivative as a Necessary Condition for a Local Extremum
- Proposition: Zorn's Lemma is Equivalent To the Axiom of Choice
- Proposition: \(\epsilon\)-\(\delta\) Definition of Continuity
- Proposition: `$0$` Is Less Than `$1$` In Ordered Fields
- Proposition: `\((xy)^{-1}=x^{-1}y^{-1}\)`
- Proposition: `\(-(x+y)=-x-y\)`
- Proposition: `\(\exp(0)=1\)`
- Proposition: `\(\exp(0)=1\)` (Complex Case)
- Proposition: `\(b\)`-Adic Fractions Are Real Cauchy Sequences
- Proposition: n-th Roots of Unity
- Proposition: p-Norm, Taxicab Norm, Euclidean Norm, Maximum Norm
- Section: Addition of Matrices and Vectors
- Section: Affine Maps
- Section: Age and Kinship Puzzles
- Section: Algebraic Structures of Complete Residue Systems
- Section: Algebraic Structures of Reduced Residue Systems
- Section: An Application of the Möbius Inversion Formula
- Section: Analytic Continuation
- Section: Asymptotic Notation
- Section: Average Case
- Section: Axioms of Connection and Their Consequences
- Section: Axioms of Order and Their Consequences
- Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines
- Section: Book 02: Fundamentals of Geometric Algebra
- Section: Book 03: Fundamentals of Plane Geometry Involving Circles
- Section: Book 04: Circles: Inscription and Circumscription
- Section: Book 05: Proportion
- Section: Book 06: Similar Figures
- Section: Book 07: Elementary Number Theory
- Section: Book 08: Continued Proportion
- Section: Book 09: Applications of Number Theory
- Section: Book 10: Incommensurable Magnitudes
- Section: Book 11: Elementary Stereometry
- Section: Book 12: Proportional Stereometry
- Section: Book 13: Platonic Solids
- Section: Boundary Value Problems
- Section: Brouwer's Theorem
- Section: Calculating Legendre Symbols
- Section: Clock Puzzles
- Section: Closed Formulas for Infinite Real Series
- Section: Closed Formulas for Sums
- Section: Common Types of Functions
- Section: Complex Cyclometric Functions
- Section: Complex Hyporbolic Functions
- Section: Complex Inverse Hyperbolic Functions
- Section: Complex Logarithms
- Section: Complex Mixed Functions
- Section: Complex Rational Functions
- Section: Complex Trigonometric Functions
- Section: Convergence and Divergence Criteria for Real Series
- Section: Cross Product
- Section: Cyclometric Functions
- Section: Differentiable Complex Functions
- Section: Differentiable Functions
- Section: Digital Puzzles
- Section: Dirichlet Convolution
- Section: Dissection Puzzles
- Section: Dual Space
- Section: Dynamical Chess Puzzles
- Section: Elimination
- Section: Examples of Limit Calculations
- Section: Examples of Propositions With Different Syntactic Forms but the Same Boolean Function
- Section: Extremal Values with Side Conditions
- Section: Fermat Numbers
- Section: Fibonacci Numbers
- Section: Generalizations of the Legendre symbol - Jacobi and Kronecker Symbols
- Section: Greek Cross Puzzles
- Section: Holomorphic Functions
- Section: Homology and Winding Numbers
- Section: Homotopy
- Section: Hyporbolic Functions
- Section: Ideals
- Section: Implicit Functions
- Section: Important Properties of Functions
- Section: Integrable Complex Functions
- Section: Integrable Functions
- Section: Inverse Hyperbolic Functions
- Section: Invertible Complex Functions
- Section: Isogonality
- Section: Iterative Methods
- Section: Iterative Methods
- Section: Jordan Measurability
- Section: Kakutani's Theorem
- Section: Linear 1. Order DE
- Section: Linear 1. Order DE
- Section: Linear 2. Order DE
- Section: Linear 2. Order DE
- Section: Linear Systems of 1. Order DE
- Section: Linear n-th Order DE
- Section: Linear vs. Non-Linear DE
- Section: Locomotion and Speed Puzzles
- Section: Logarithms
- Section: Magic Squares of Primes
- Section: Mixed Functions
- Section: Moebius Transformations
- Section: Money Puzzles
- Section: Multi-Step Methods
- Section: Multiple Lebesgue Integrals
- Section: Multiple Riemann Integrals
- Section: NP-Completeness
- Section: Non-Linear DE
- Section: Normal Forms
- Section: One-Step Methods
- Section: Order of DE
- Section: Patchwork Puzzles
- Section: Path Integrals
- Section: Paths
- Section: Periodic Complex Functions
- Section: Polynomial Time
- Section: Probabilistic Complexity
- Section: Quadrics
- Section: Real Intervals and Bounded Real Sets
- Section: Riemann's Plane and Sphere
- Section: Schauder's Theorem
- Section: Solutions of Polynomials
- Section: Solving General Systems Of Linear Equations - Gaussian Method
- Section: Solving Simple Systems of Linear Equations
- Section: Some Properties of the Möbius Function
- Section: Statical Chess Puzzles
- Section: Subspaces
- Section: Subtracting, Multiplying, and Dividing Magics
- Section: Systems of DE
- Section: The Chessboard
- Section: The Guarded Chessboard
- Section: Theorems Regarding Limits Of Sequences
- Section: Theorems Regarding Limits of Functions
- Section: Trigonometric Functions
- Section: Turing Machine
- Section: Uniform Complexity Measure and Bit Complexity
- Section: Uniform Convergence of Functions
- Section: Various Arithmetical and Algebraic Problems
- Section: Various Chess Puzzles
- Section: Various Dissection Puzzles
- Section: Various Geometrical Puzzles
- Section: Worst Case
- Solution: (related to Problem: Verifying Group Properties)
- Solution: (related to Problem: Verifying Subgroup Properties)
- Solution: (related to Problem: Calculating an Infinite Sum)
- Solution: (related to Problem: Interpolating Numbers With a Polynomial)
- Solution: (related to Problem: Sum of Consecutive Positive Integers)
- Solution: (related to Problem: Sum of Consecutive Squares)
- Solution: (related to Problem: Sums of Falling Factorial Powers)
- Solution: (related to Problem: Sums of Powers of Two)
- Solution: (related to Problem: A Census Puzzle)
- Solution: (related to Problem: A Family Party)
- Solution: (related to Problem: A Mixed Pedigree)
- Solution: (related to Problem: Concerning Tommy's Age)
- Solution: (related to Problem: Heard on the Tube Railway)
- Solution: (related to Problem: How Old Was Mary?)
- Solution: (related to Problem: Mamma's Age)
- Solution: (related to Problem: Mary and Marmaduke)
- Solution: (related to Problem: Mother and Daughter)
- Solution: (related to Problem: Mrs. Timpkin's Age)
- Solution: (related to Problem: Next-door Neighbors)
- Solution: (related to Problem: Queer Relationships)
- Solution: (related to Problem: Rover's Age)
- Solution: (related to Problem: The Bag of Nuts)
- Solution: (related to Problem: The Family Ages)
- Solution: (related to Problem: Their Ages)
- Solution: (related to Problem: Wilson's Poser)
- Solution: (related to Problem: A Puzzling Watch)
- Solution: (related to Problem: A Time Puzzle)
- Solution: (related to Problem: Changing Places)
- Solution: (related to Problem: The Club Clock)
- Solution: (related to Problem: The Railway Station Clock)
- Solution: (related to Problem: The Stop-Watch)
- Solution: (related to Problem: The Three Clocks)
- Solution: (related to Problem: The Village Simpleton)
- Solution: (related to Problem: The Wapshaw's Wharf Mystery)
- Solution: (related to Problem: What Was the Time?)
- Solution: (related to Problem: Adding The Digits)
- Solution: (related to Problem: Digital Division)
- Solution: (related to Problem: Digital Multiplication)
- Solution: (related to Problem: Digital Square Numbers)
- Solution: (related to Problem: Digits and Squares)
- Solution: (related to Problem: More Mixed Fractions)
- Solution: (related to Problem: Odd And Even Digits)
- Solution: (related to Problem: Queer Multiplication)
- Solution: (related to Problem: The Barrel of Beer)
- Solution: (related to Problem: The Cab Numbers)
- Solution: (related to Problem: The Century Puzzle)
- Solution: (related to Problem: The Dice Numbers)
- Solution: (related to Problem: The Digital Century)
- Solution: (related to Problem: The Four Sevens)
- Solution: (related to Problem: The Lockers Puzzle)
- Solution: (related to Problem: The Mystic Eleven)
- Solution: (related to Problem: The Nine Counters)
- Solution: (related to Problem: The Number Checks Puzzle)
- Solution: (related to Problem: The Pierrot's Puzzle)
- Solution: (related to Problem: The Ten Counters)
- Solution: (related to Problem: The Three Groups)
- Solution: (related to Problem: Average Speed)
- Solution: (related to Problem: Donkey Riding)
- Solution: (related to Problem: Drawing Her Pension)
- Solution: (related to Problem: Sir Edwyn De Tudor)
- Solution: (related to Problem: The Basket Of Potatoes)
- Solution: (related to Problem: The Hydroplane Question)
- Solution: (related to Problem: The Passenger's Fare)
- Solution: (related to Problem: The Three Villages)
- Solution: (related to Problem: The Two Trains)
- Solution: (related to Problem: A Charitable Bequest)
- Solution: (related to Problem: A Deal in Apples)
- Solution: (related to Problem: A Deal in Eggs)
- Solution: (related to Problem: A New Money Puzzle)
- Solution: (related to Problem: A Postoffice Perplexity)
- Solution: (related to Problem: A Puzzle in Reversals)
- Solution: (related to Problem: A Queer Coincidence)
- Solution: (related to Problem: A Queer Thing in Money)
- Solution: (related to Problem: A Shopping Perplexity)
- Solution: (related to Problem: At a Cattle Market)
- Solution: (related to Problem: Awkward Money)
- Solution: (related to Problem: Beef and Sausages)
- Solution: (related to Problem: Buying Apples)
- Solution: (related to Problem: Buying Chestnuts)
- Solution: (related to Problem: Buying Presents)
- Solution: (related to Problem: Defective Observation)
- Solution: (related to Problem: Domestic Economy)
- Solution: (related to Problem: Giving Change)
- Solution: (related to Problem: Indiscriminate Charity)
- Solution: (related to Problem: Judkins's Cattle)
- Solution: (related to Problem: Pocket Money)
- Solution: (related to Problem: Square Money)
- Solution: (related to Problem: The Beanfeast Puzzle)
- Solution: (related to Problem: The Bicylce Thief)
- Solution: (related to Problem: The Broken Coins)
- Solution: (related to Problem: The Christmas-Boxes)
- Solution: (related to Problem: The Costermonger's Puzzle)
- Solution: (related to Problem: The Cyclists' Feast)
- Solution: (related to Problem: The Excursion Ticket Puzzle)
- Solution: (related to Problem: The Grocer and Draper)
- Solution: (related to Problem: The Junior Clerk's Puzzle)
- Solution: (related to Problem: The Market Women)
- Solution: (related to Problem: The Millionaire's Perplexity)
- Solution: (related to Problem: The Puzzling Money-Boxes)
- Solution: (related to Problem: The Two Aeroplanes)
- Solution: (related to Problem: The Widow's Legacy)
- Solution: (related to Problem: The new Year's Eve Suppers)
- Solution: (related to Problem: Two Questions in Probabilities)
- Solution: (related to Problem: Youthful Precocity)
- Solution: (related to Problem: A Fence Problem)
- Solution: (related to Problem: A Legal Difficulty)
- Solution: (related to Problem: A Printer's Error)
- Solution: (related to Problem: A Problem in Squares)
- Solution: (related to Problem: A Puzzling Legacy)
- Solution: (related to Problem: A Question of Definition)
- Solution: (related to Problem: A Study in Thrift)
- Solution: (related to Problem: Academic Courtesies)
- Solution: (related to Problem: Catching the Thief)
- Solution: (related to Problem: Circling the Squares)
- Solution: (related to Problem: Curious Numbers)
- Solution: (related to Problem: Find Ada's Surname)
- Solution: (related to Problem: Heads or Tails)
- Solution: (related to Problem: Mr. Gubbins in a Fog)
- Solution: (related to Problem: Painting the Lamp-Posts)
- Solution: (related to Problem: Rackbrane's Little Loss)
- Solution: (related to Problem: Reeping the Corn)
- Solution: (related to Problem: Saturday Marketing)
- Solution: (related to Problem: Simple Division)
- Solution: (related to Problem: Simple Multiplication)
- Solution: (related to Problem: The Abbot's Puzzle)
- Solution: (related to Problem: The Artilleryman's Dilemma)
- Solution: (related to Problem: The Banker's Puzzle)
- Solution: (related to Problem: The Battle of Hastings)
- Solution: (related to Problem: The Converted Miser)
- Solution: (related to Problem: The Dutchmen's Wives)
- Solution: (related to Problem: The Farmer and His Sheep)
- Solution: (related to Problem: The Five Brigands)
- Solution: (related to Problem: The Great Scramble)
- Solution: (related to Problem: The Laborer's Puzzle)
- Solution: (related to Problem: The Leap-Year Ladies)
- Solution: (related to Problem: The Miners' Holiday)
- Solution: (related to Problem: The Muddletown Election)
- Solution: (related to Problem: The Nine Treasure Boxes)
- Solution: (related to Problem: The Parish Council Election)
- Solution: (related to Problem: The Sculptor's Problem)
- Solution: (related to Problem: The See-saw Puzzle)
- Solution: (related to Problem: The Spanish Miser)
- Solution: (related to Problem: The Spot on the Table)
- Solution: (related to Problem: The Stonemason's Problem)
- Solution: (related to Problem: The Suffragists' Meeting)
- Solution: (related to Problem: The Sultan's Army)
- Solution: (related to Problem: The Thirty-Three Pearls)
- Solution: (related to Problem: The Torn Number)
- Solution: (related to Problem: The Trusses of Hay)
- Solution: (related to Problem: A Dungeon Puzzle)
- Solution: (related to Problem: A New Bishop's Puzzle)
- Solution: (related to Problem: A New Counter Puzzle)
- Solution: (related to Problem: An Episcopal Visitation)
- Solution: (related to Problem: Exercise For Prisoners)
- Solution: (related to Problem: Farmer Lawrence's Cornfields)
- Solution: (related to Problem: St. George And The Dragon)
- Solution: (related to Problem: The Board In Compartments)
- Solution: (related to Problem: The Cubic Knight's Tour)
- Solution: (related to Problem: The Forty-nine Stars)
- Solution: (related to Problem: The Four Frogs)
- Solution: (related to Problem: The Four Kangaroos)
- Solution: (related to Problem: The Four Knights' Tours)
- Solution: (related to Problem: The Greyhound Puzzle)
- Solution: (related to Problem: The Kennel Puzzle)
- Solution: (related to Problem: The Languishing Maiden)
- Solution: (related to Problem: The Lion And The Man)
- Solution: (related to Problem: The Mandarin's Puzzle)
- Solution: (related to Problem: The Queen's Journey)
- Solution: (related to Problem: The Queen's Tour)
- Solution: (related to Problem: The Rook's Journey)
- Solution: (related to Problem: The Rook's Tour)
- Solution: (related to Problem: The Scientific Skater)
- Solution: (related to Problem: The Star Puzzle)
- Solution: (related to Problem: The Two Pawns)
- Solution: (related to Problem: The Yacht Race)
- Solution: (related to Problem: A Problem in Mosaics)
- Solution: (related to Problem: A Puzzle With Pawns)
- Solution: (related to Problem: Bachet's Square)
- Solution: (related to Problem: Bishops in Convocation)
- Solution: (related to Problem: Bishops — guarded)
- Solution: (related to Problem: Bishops — unguarded)
- Solution: (related to Problem: Lion-hunting)
- Solution: (related to Problem: Queens And Bishop Puzzle)
- Solution: (related to Problem: The Amazons)
- Solution: (related to Problem: The Coloured Counters)
- Solution: (related to Problem: The Crowded Chessboard)
- Solution: (related to Problem: The Eight Queens)
- Solution: (related to Problem: The Eight Rooks)
- Solution: (related to Problem: The Eight Stars)
- Solution: (related to Problem: The Five Crescents of Byzantium)
- Solution: (related to Problem: The Five Dogs Puzzle)
- Solution: (related to Problem: The Forty-nine Counters)
- Solution: (related to Problem: The Four Lions)
- Solution: (related to Problem: The Gentle Art Of Stamp-licking)
- Solution: (related to Problem: The Hat-peg Puzzle)
- Solution: (related to Problem: The Knight-guards)
- Solution: (related to Problem: The Southern Cross)
- Solution: (related to Problem: The Thirty-six Letter Blocks)
- Solution: (related to Problem: The Three Sheep)
- Solution: (related to Problem: Under the Veil)
- Solution: (related to Problem: Boards With An Odd Number Of Squares)
- Solution: (related to Problem: Chequered Board Divisions)
- Solution: (related to Problem: Lions And Crowns)
- Solution: (related to Problem: The Abbot's Window)
- Solution: (related to Problem: The Chessboard Sentence)
- Solution: (related to Problem: The Chinese Chessboard)
- Solution: (related to Problem: The Grand Lama's Problem)
- Solution: (related to Problem: An Amazing Dilemma)
- Solution: (related to Problem: Ancient Chinese Puzzle)
- Solution: (related to Problem: Checkmate!)
- Solution: (related to Problem: Chessboard Solitaire)
- Solution: (related to Problem: Counter Solitaire)
- Solution: (related to Problem: Counting The Rectangles)
- Solution: (related to Problem: Immovable Pawns)
- Solution: (related to Problem: Queer Chess)
- Solution: (related to Problem: Setting The Board)
- Solution: (related to Problem: Stalemate)
- Solution: (related to Problem: The Crusader)
- Solution: (related to Problem: The Forsaken King)
- Solution: (related to Problem: The Monstrosity)
- Solution: (related to Problem: The Rookery)
- Solution: (related to Problem: The Six Pawns)
- Solution: (related to Problem: Thirty-six Mates)
- Solution: (related to Problem: A Dormitory Puzzle)
- Solution: (related to Problem: A Puzzle for Card-players)
- Solution: (related to Problem: A Tennis Tournament)
- Solution: (related to Problem: An Acrostic Puzzle)
- Solution: (related to Problem: Building The Tetrahedron)
- Solution: (related to Problem: Counter Crosses)
- Solution: (related to Problem: Fifteen Letter Puzzle)
- Solution: (related to Problem: King Arthur's Knights)
- Solution: (related to Problem: Painting A Pyramid)
- Solution: (related to Problem: Painting The Die)
- Solution: (related to Problem: The Antiquary's Chain)
- Solution: (related to Problem: The Barrels Of Balsam)
- Solution: (related to Problem: The City Luncheons)
- Solution: (related to Problem: The Cross Target)
- Solution: (related to Problem: The Eight Villas)
- Solution: (related to Problem: The Fifteen Dominoes)
- Solution: (related to Problem: The Four Postage Stamps)
- Solution: (related to Problem: The Glass Balls)
- Solution: (related to Problem: The Mouse-trap Puzzle)
- Solution: (related to Problem: The Nine Schoolboys)
- Solution: (related to Problem: The Peal Of Bells)
- Solution: (related to Problem: The Round Table)
- Solution: (related to Problem: The Sixteen Sheep)
- Solution: (related to Problem: The Wrong Hats)
- Solution: (related to Problem: Those Fifteen Sheep)
- Solution: (related to Problem: Three Men In A Boat)
- Solution: (related to Problem: Crossing The River Axe)
- Solution: (related to Problem: Crossing The Stream)
- Solution: (related to Problem: Five Jealous Husbands)
- Solution: (related to Problem: Stealing The Castle Treasure)
- Solution: (related to Problem: The Four Elopements)
- Solution: (related to Problem: The Cross and the Triangle)
- Solution: (related to Problem: The Folded Cross)
- Solution: (related to Problem: The Silk Patchwork)
- Solution: (related to Problem: Two Crosses From One)
- Solution: (related to Problem: Another Linoleum Puzzle)
- Solution: (related to Problem: Another Patchwork Puzzle)
- Solution: (related to Problem: He Banner Puzzle)
- Solution: (related to Problem: Linoleum Cutting)
- Solution: (related to Problem: Mrs. Perkins's Quilt)
- Solution: (related to Problem: Mrs. Smiley's Christmas Present)
- Solution: (related to Problem: The Cushion Covers)
- Solution: (related to Problem: The Squares Of Brocade)
- Solution: (related to Problem: A Cutting-out Puzzle)
- Solution: (related to Problem: A Tangram Paradox)
- Solution: (related to Problem: An Easy Dissection Puzzle)
- Solution: (related to Problem: An Easy Square Puzzle)
- Solution: (related to Problem: Another Joiner's Problem)
- Solution: (related to Problem: Dissecting a Mittre)
- Solution: (related to Problem: Mrs. Hobson's Hearthrug)
- Solution: (related to Problem: The Betsy Ross Puzzle)
- Solution: (related to Problem: The Bun Puzzle)
- Solution: (related to Problem: The Cardboard Chain)
- Solution: (related to Problem: The Chocolate Squares)
- Solution: (related to Problem: The Christmas Pudding)
- Solution: (related to Problem: The Dissected Triangle)
- Solution: (related to Problem: The Great Monad)
- Solution: (related to Problem: The Joiner's Problem)
- Solution: (related to Problem: The Landowner's Fences)
- Solution: (related to Problem: The Pentagon and Square)
- Solution: (related to Problem: The Potato Puzzle)
- Solution: (related to Problem: The Seven Pigs)
- Solution: (related to Problem: The Square of Veneer)
- Solution: (related to Problem: The Table-top and Stools)
- Solution: (related to Problem: The Two Horseshoes)
- Solution: (related to Problem: The Wizard's Cats)
- Solution: (related to Problem: A Kite-flying Puzzle)
- Solution: (related to Problem: A New Match Puzzle)
- Solution: (related to Problem: Concerning Wheels)
- Solution: (related to Problem: Drawing A Spiral)
- Solution: (related to Problem: Farmer Wurzel's Estate)
- Solution: (related to Problem: How To Draw An Oval)
- Solution: (related to Problem: How To Make Cisterns)
- Solution: (related to Problem: Lady Belinda's Garden)
- Solution: (related to Problem: Papa's Puzzle)
- Solution: (related to Problem: St. George's Banner)
- Solution: (related to Problem: Stealing The Bell-ropes)
- Solution: (related to Problem: The Ball Problem)
- Solution: (related to Problem: The Cardboard Box)
- Solution: (related to Problem: The Clothes Line Puzzle)
- Solution: (related to Problem: The Compasses Puzzle)
- Solution: (related to Problem: The Cone Puzzle)
- Solution: (related to Problem: The Crescent Puzzle)
- Solution: (related to Problem: The Eight Sticks)
- Solution: (related to Problem: The Four Sons)
- Solution: (related to Problem: The Garden Puzzle)
- Solution: (related to Problem: The Garden Walls)
- Solution: (related to Problem: The Milkmaid Puzzle)
- Solution: (related to Problem: The Puzzle Wall)
- Solution: (related to Problem: The Sheep-fold)
- Solution: (related to Problem: The Six Sheep-pens)
- Solution: (related to Problem: The Tethered Goat)
- Solution: (related to Problem: The Three Railway Stations)
- Solution: (related to Problem: The Yorkshire Estates)
- Solution: (related to Problem: Card Magic Squares)
- Solution: (related to Problem: Eight Jolly Gaol Birds)
- Solution: (related to Problem: Nine Jolly Gaol Birds)
- Solution: (related to Problem: The Eighteen Dominoes)
- Solution: (related to Problem: The Magic Strips)
- Solution: (related to Problem: The Siberian Dungeons)
- Solution: (related to Problem: The Spanish Dungeon)
- Solution: (related to Problem: The Troublesome Eight)
- Solution: (related to Problem: Magic Squares Of Two Degrees)
- Solution: (related to Problem: Two New Magic Squares)
- Solution: (related to Problem: A Magic Square Of Composites)
- Solution: (related to Problem: The Baskets Of Plums)
- Solution: (related to Problem: The Magic Knight's Tour)
- Solution: (related to Problem: The Mandarin's "T" Puzzle)
- Solution: (related to Problem: A Packing Puzzle)
- Solution: (related to Problem: Gold Packing In Russia)
- Solution: (related to Problem: Mixing The Tea)
- Solution: (related to Problem: New Measuring Puzzle)
- Solution: (related to Problem: The Barrel Puzzle)
- Solution: (related to Problem: The Barrels Of Honey)
- Solution: (related to Problem: The Doctor's Query)
- Solution: (related to Problem: The Honest Dairyman)
- Solution: (related to Problem: The Keg Of Wine)
- Solution: (related to Problem: The Wassail Bowl)
- Solution: (related to Problem: Wine And Water)
- Solution: (related to Problem: A Lodging-house Difficulty)
- Solution: (related to Problem: A Railway Muddle)
- Solution: (related to Problem: A Railway Puzzle)
- Solution: (related to Problem: Arranging The Jampots)
- Solution: (related to Problem: Boys And Girls)
- Solution: (related to Problem: Catching The Mice)
- Solution: (related to Problem: Central Solitaire)
- Solution: (related to Problem: Plates And Coins)
- Solution: (related to Problem: Round The Coast)
- Solution: (related to Problem: The Eccentric Cheesemonger)
- Solution: (related to Problem: The Educated Frogs)
- Solution: (related to Problem: The Eight Engines)
- Solution: (related to Problem: The Exchange Puzzle)
- Solution: (related to Problem: The Grasshopper Puzzle)
- Solution: (related to Problem: The Hat Puzzle)
- Solution: (related to Problem: The Letter Block Puzzle)
- Solution: (related to Problem: The Motor-garage Puzzle)
- Solution: (related to Problem: The Nine Almonds)
- Solution: (related to Problem: The Six Frogs)
- Solution: (related to Problem: The Ten Apples)
- Solution: (related to Problem: The Ten Prisoners)
- Solution: (related to Problem: The Twelve Pennies)
- Solution: (related to Problem: The Twickenham Puzzle)
- Solution: (related to Problem: The Victoria Cross Puzzle)
- Solution: (related to Problem: Torpedo Practice)
- Solution: (related to Problem: A Plantation Puzzle)
- Solution: (related to Problem: Cherries And Plums)
- Solution: (related to Problem: The Burmese Plantation)
- Solution: (related to Problem: The King And The Castles)
- Solution: (related to Problem: The Ten Coins)
- Solution: (related to Problem: The Twelve Mince-pies)
- Solution: (related to Problem: The Twenty-one Trees)
- Solution: (related to Problem: Turks And Russians)
- Solution: (related to Problem: "Strand" Patience)
- Solution: (related to Problem: A Trick With Dice)
- Solution: (related to Problem: Card Triangles)
- Solution: (related to Problem: Dominoes In Progression)
- Solution: (related to Problem: Slow Cricket)
- Solution: (related to Problem: The "T" Card Puzzle)
- Solution: (related to Problem: The Card Frame Puzzle)
- Solution: (related to Problem: The Cross Of Cards)
- Solution: (related to Problem: The Domino Frame Puzzle)
- Solution: (related to Problem: The Five Dominoes)
- Solution: (related to Problem: The Football Players)
- Solution: (related to Problem: The Horse-race Puzzle)
- Solution: (related to Problem: The Motor-car Race)
- Solution: (related to Problem: The Village Cricket Match)
- Solution: (related to Problem: A Match Mystery)
- Solution: (related to Problem: A War Puzzle Game)
- Solution: (related to Problem: Puss In The Corner)
- Solution: (related to Problem: The Cigar Puzzle)
- Solution: (related to Problem: The Montenegrin Dice Game)
- Solution: (related to Problem: The Pebble Game)
- Solution: (related to Problem: The Two Rooks)
- Solution: (related to Problem: A Chessboard Fallacy)
- Solution: (related to Problem: A Calendar Puzzle)
- Solution: (related to Problem: A Chain Puzzle)
- Solution: (related to Problem: A Wonderful Village)
- Solution: (related to Problem: Find The Man's Wife)
- Solution: (related to Problem: Jack And The Beanstalk)
- Solution: (related to Problem: Pheasant-shooting)
- Solution: (related to Problem: Placing Halfpennies)
- Solution: (related to Problem: Such A Getting Upstairs)
- Solution: (related to Problem: The Dovetailed Block)
- Solution: (related to Problem: The Five Pennies)
- Solution: (related to Problem: The Gardener And The Cook)
- Solution: (related to Problem: The Hymn-board Poser)
- Solution: (related to Problem: The Industrious Bookworm)
- Solution: (related to Problem: The Ruby Brooch)
- Solution: (related to Problem: The Sabbath Puzzle)
- Solution: (related to Problem: The Tiring Irons)
- Solution: (related to Problem: Who Was First?)
- Solution: (related to Problem: A Bank Holiday Puzzle)
- Solution: (related to Problem: A Juvenile Puzzle)
- Solution: (related to Problem: A Puzzle For Motorists)
- Solution: (related to Problem: Hannah's Puzzle)
- Solution: (related to Problem: Inspecting A Mine)
- Solution: (related to Problem: The Cyclists' Tour)
- Solution: (related to Problem: The Deified Puzzle)
- Solution: (related to Problem: The Diamond Puzzle)
- Solution: (related to Problem: The Dissected Circle)
- Solution: (related to Problem: The Fifteen Turnings)
- Solution: (related to Problem: The Fly On The Octahedron)
- Solution: (related to Problem: The Grand Tour)
- Solution: (related to Problem: The Honeycomb Puzzle)
- Solution: (related to Problem: The Icosahedron Puzzle)
- Solution: (related to Problem: The Level Puzzle)
- Solution: (related to Problem: The Monk And The Bridges)
- Solution: (related to Problem: The Motor-car Tour)
- Solution: (related to Problem: The Sailor's Puzzle)
- Solution: (related to Problem: The Tube Inspector's Puzzle)
- Solution: (related to Problem: The Union Jack)
- Solution: (related to Problem: The Voters' Puzzle)
- Solution: (related to Problem: Visiting The Towns)
- Solution: (related to Problem: Water, Gas, And Electricity)
- Solution: (related to Problem: Checking if `$K_5$` is Planar)
- Solution: (related to Problem: Checking if `$K_{3,3}$` is Planar)
- Solution: (related to Problem: Applications of the Jacobi Symbol)
- Solution: (related to Problem: Calculating Quadratic Residues)
- Solution: (related to Problem: To Construct a Partition of a Given Set)
- Solution: Application of the Law of Total Probability (related to Problem: Broken Items in the Box)
- Solution: Application of the Urn Model Without Replacement (related to Problem: Broken Items in the Box II)
- Solution: The Six Frogs - Solved (related to Problem: The Six Frogs)
- Subsection: Axioms from Book 1
- Subsection: Common Notions (all Books)
- Subsection: Convergence Criteria for Infinite Series Involving Products
- Subsection: Criteria for Holomorphic Functions
- Subsection: Cross-Ratio
- Subsection: Definitions From Book 10 (II)
- Subsection: Definitions From Book 10 (III)
- Subsection: Definitions from Book 1
- Subsection: Definitions from Book 10 (I)
- Subsection: Definitions from Book 11
- Subsection: Definitions from Book 2
- Subsection: Definitions from Book 3
- Subsection: Definitions from Book 4
- Subsection: Definitions from Book 5
- Subsection: Definitions from Book 6
- Subsection: Definitions from Book 7
- Subsection: Extended Concept of the Riemann Integral - the Improper Integral
- Subsection: Fixed Points
- Subsection: Harmonic Functions
- Subsection: Indefinite Integral
- Subsection: Inversion
- Subsection: Path Integrals
- Subsection: Properties
- Subsection: Propositions from Book 1
- Subsection: Propositions from Book 10
- Subsection: Propositions from Book 11
- Subsection: Propositions from Book 12
- Subsection: Propositions from Book 13
- Subsection: Propositions from Book 2
- Subsection: Propositions from Book 3
- Subsection: Propositions from Book 4
- Subsection: Propositions from Book 5
- Subsection: Propositions from Book 6
- Subsection: Propositions from Book 7
- Subsection: Propositions from Book 8
- Subsection: Propositions from Book 9
- Subsection: Riemann Integral
- Subsection: Riemann-Stieltjes Integral
- Theorem: Approximation of Factorials Using the Stirling Formula
- Theorem: Bayes' Theorem
- Theorem: Bernoulli's Inequality
- Theorem: Binomial Theorem
- Theorem: Brooks' Theorem
- Theorem: Characterization of Biconnected Planar Graphs
- Theorem: Characterization of Bipartite Graphs
- Theorem: Characterization of Eulerian Graphs
- Theorem: Characterization of Planar Graphs
- Theorem: Characterization of Planar Hamiltonian Graphs
- Theorem: Characterization of Semi-Eulerian Graphs
- Theorem: Chinese Remainder Theorem
- Theorem: Classification of Cyclic Groups
- Theorem: Classification of Finite Groups with the Order of a Prime Number
- Theorem: Commutative Group of Multiplicative Functions
- Theorem: Completeness Principle for Complex Numbers
- Theorem: Completeness Principle for Real Numbers
- Theorem: Connection between Rings, Ideals, and Fields
- Theorem: Construction of Fields from Integral Domains
- Theorem: Construction of Groups from Commutative and Cancellative Semigroups
- Theorem: Continuous Functions Mapping Compact Domains to Real Numbers Take Maximum and Minimum Values on these Domains
- Theorem: Continuous Functions on Compact Domains are Uniformly Continuous
- Theorem: Continuous Real Functions on Closed Intervals are Uniformly Continuous
- Theorem: Darboux's Theorem
- Theorem: De Moivre's Identity, Complex Powers
- Theorem: Defining Properties of the Field of Real Numbers
- Theorem: Deterministic Finite Automata Describe Regular Languages
- Theorem: Distinction Between Finite and Infinite Sets Using Subsets
- Theorem: Euler Characteristic for Planar Graphs
- Theorem: Euler-Fermat Theorem
- Theorem: Every Bounded Monotonic Sequence Is Convergent
- Theorem: Every Bounded Real Sequence has a Convergent Subsequence
- Theorem: Existence and Uniqueness of a Straight Line Through Two Points
- Theorem: Fermat's Last Theorem
- Theorem: Finite Basis Theorem
- Theorem: Finite Integral Domains are Fields
- Theorem: First Isomorphism Theorem for Groups
- Theorem: First Law of Planetary Motion
- Theorem: First Supplementary Law to the Quadratic Reciprocity Law
- Theorem: Five Color Theorem for Planar Graphs
- Theorem: Four Color Theorem for Planar Graphs
- Theorem: Four Color Theorem for Planar Graphs With a Dual Hamiltonian Graph
- Theorem: Fundamental Theorem of Arithmetic
- Theorem: Fundamental Theorem of Calculus
- Theorem: Fundamental Theorem of the Difference Calculus
- Theorem: Heine-Borel Theorem
- Theorem: Inclusion-Exclusion Principle (Sylvester's Formula)
- Theorem: Indefinite Integral, Antiderivative
- Theorem: Inequality Between the Geometric and the Arithmetic Mean
- Theorem: Inequality of Weighted Arithmetic Mean
- Theorem: Inequality of the Arithmetic Mean
- Theorem: Infinite Set of Prime Numbers
- Theorem: Integration by Substitution
- Theorem: Intermediate Root Value Theorem
- Theorem: Intermediate Value Theorem
- Theorem: Isomorphism of Rings
- Theorem: Law of Total Probability
- Theorem: Mean Value Theorem For Riemann Integrals
- Theorem: Mostowski's Theorem
- Theorem: Multinomial Theorem
- Theorem: Möbius Inversion Formula
- Theorem: Nested Closed Subset Theorem
- Theorem: Number of Labeled Spanning Trees
- Theorem: Number of Multiples of a Prime Number Less Than Factorial
- Theorem: Order of Cyclic Group (Fermat's Little Theorem)
- Theorem: Order of Subgroup Divides Order of Finite Group
- Theorem: Partial Integration
- Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic
- Theorem: Properties of the Jacobi Symbol
- Theorem: Quadratic Reciprocity Law
- Theorem: Reduction of NFA to DFA (Rabin-Scott Theorem)
- Theorem: Reduction of `$\epsilon$`-NFA to NFA
- Theorem: Relationship Between the Solutions of Homogeneous and Inhomogeneous SLEs
- Theorem: Reverse Triangle Inequalities
- Theorem: Rolle's Theorem
- Theorem: Schröder-Bernstein Theorem
- Theorem: Second Law of Planetary Motion
- Theorem: Second Supplementary Law to the Quadratic Reciprocity Law
- Theorem: Simulating LOOP Programs Using WHILE Programs
- Theorem: Simulating WHILE Programs Using GOTO Programs (and vice versa)
- Theorem: Six Color Theorem for Planar Graphs
- Theorem: Squeezing Theorem for Functions
- Theorem: Supremum Property, Infimum Property
- Theorem: Taylor's Formula
- Theorem: Taylor's Formula Using the Difference Operator
- Theorem: Theorem of Bolzano-Weierstrass
- Theorem: Theorem of Large Numbers for Relative Frequencies
- Theorem: Third Law of Planetary Motion
- Theorem: Triangle Inequality
- Theorem: Trichotomy of Ordinals
- Topic: Ancient Chinese Mathematics
- Topic: Babylonian Mathematics
- Topic: Egyptian Mathematics
- Topic: Indian Mathematics
- Topic: Sumerian Mathematics
- Topic: to 22 000 BC: Ishango Bone
- Topic: to 4000 BC: Clay Accounting Tokens
From non-Github
- @A-Alberola (1)
- @A-Cervi (1)
- @A-D-Irvine (1)
- @Alex-D-D-Craik (5)
- @Alistair-Watson (1)
- @Andrew-Rich (2)
- @Andrew-Whitaker (1)
- @Antonia-Martinez (1)
- @Benjamin-Baumslag (1)
- @Brenner (99)
- Axiom: Archimedean Axiom
- Corollary: Properties of a Real Scalar Product (related to Definition: Dot Product, Inner Product, Scalar Product (General Field Case))
- Definition: (Unit) Ring
- Definition: Algebra over a Ring
- Definition: Algebraic Element
- Definition: Alternating Multilinear Map
- Definition: Bilinear Form
- Definition: Binary Operation
- Definition: Bounded Subset of a Metric Space
- Definition: Canonical Projection
- Definition: Commutative (Abelian) Group
- Definition: Commutative (Unit) Ring
- Definition: Comparing the Elements of Posets and Chains
- Definition: Complete Ordered Field
- Definition: Constant Function
- Definition: Cotangent Bundle
- Definition: Derivative of an n-Dimensional Curve
- Definition: Differentiable Manifold, Atlas
- Definition: Differential Form of Degree k
- Definition: Direct Sum of Vector Spaces
- Definition: Directional Derivative
- Definition: Eigenvalue
- Definition: Eigenvector
- Definition: Embedding, Inclusion Map
- Definition: Endomorphism
- Definition: Exterior Algebra, Alternating Product, Universal Alternating Map
- Definition: Field
- Definition: Field Extension
- Definition: Finite Field Extension
- Definition: Finite and Sigma-Finite Measure
- Definition: Finite and Sigma-Finite Pre-measure
- Definition: First-Order Ordinary Differential Equation (ODE)
- Definition: Generalized Polynomial Function
- Definition: Group
- Definition: Group Operation
- Definition: Higher Order Directional Derivative
- Definition: Homeomorphism, Homeomorphic Spaces
- Definition: Index Set and Set Family
- Definition: Integral Closure
- Definition: Integral Element
- Definition: Limit of a Function
- Definition: Linear Map
- Definition: Manifold
- Definition: Matrix, Set of Matrices over a Field
- Definition: Maximal Ideal
- Definition: Measurable Set
- Definition: Measurable Space
- Definition: Measure
- Definition: Measureable Function
- Definition: Minimal Polynomial
- Definition: Module
- Definition: Monoid
- Definition: Multilinear Map
- Definition: Multiplicative System
- Definition: Open Cover
- Definition: Open Function, Closed Function
- Definition: Ordered Field
- Definition: Partial and Total Maps (Functions)
- Definition: Polynomial Ring
- Definition: Pre-measure
- Definition: Principal Ideal
- Definition: Principal Ideal Domain
- Definition: Recursive Definition of the Determinant
- Definition: Ring Homomorphism
- Definition: Ring of Integers
- Definition: Ring of Sets (measure-theoretic definition)
- Definition: Sigma-Algebra
- Definition: Solution of Ordinary DE
- Definition: Spectrum of a Commutative Ring
- Definition: Subring
- Definition: Substructure
- Definition: Surjective Function
- Definition: Symmetric Bilinear Form
- Definition: Tangent Bundle
- Definition: Topological Chart
- Definition: Topological Space, Topology
- Definition: Total Order and Chain
- Definition: Totally Differentiable Functions, Total Derivative
- Definition: Transcendental Element
- Definition: Transition Map
- Definition: Vector Field
- Definition: Zariski Topology of a Commutative Ring
- Definition: `\(C^n\)` Differentiable Function
- Definition: `\(C^{n}\)`-Diffeomorphism
- Definition: `\(n\)` times Continuously Differentiable Functions
- Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace
- Lemma: Fiber of Maximal Ideals
- Lemma: Fiber of Prime Ideals
- Lemma: Fiber of Prime Ideals Under a Spectrum Function
- Lemma: One-to-one Correspondence of Ideals in the Factor Ring and a Commutative Ring
- Lemma: Prime Ideals of Multiplicative Systems in Integral Domains
- Proof: (related to Lemma: Equivalency of Vectors in Vector Space If their Difference Forms a Subspace)
- Proof: (related to Lemma: Fiber of Prime Ideals Under a Spectrum Function)
- Proof: (related to Proposition: Open and Closed Subsets of a Zariski Topology)
- Proof: (related to Proposition: Quotient Space)
- Proposition: Open and Closed Subsets of a Zariski Topology
- Proposition: Properties of a Complex Scalar Product
- Proposition: Quotient Space
- Proposition: Spectrum Function of Commutative Rings
- @Calahan (142)
- Axiom: 1.1: Straight Line Determined by Two Distinct Points
- Axiom: 1.2: Segment Extension
- Axiom: 1.3: Circle Determined by its Center and its Radius
- Chapter: Euclid's “Elements”
- Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle)
- Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II)
- Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III)
- Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I)
- Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines)
- Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Definition: 1.02: Line, Curve
- Definition: 1.03: Intersections of Lines
- Definition: 1.05: Surface
- Definition: 1.06: Intersections of Surfaces
- Definition: 1.09: Angle, Rectilinear, Vertex, Legs
- Definition: 1.10: Right Angle, Perpendicular Straight Lines
- Definition: 1.11: Obtuse Angle
- Definition: 1.15: Circle, Circumference, Radius
- Definition: 1.17: Diameter of the Circle
- Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure
- Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle
- Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium
- Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides
- Definition: 3.02: Tangent to the Circle, Straight Line Touching The Circle
- Definition: 3.04: Chords Equally Far From the Center of a Circle
- Definition: 3.06: Segment of a Circle, Arc
- Definition: 3.10: Circular Sector, Central Angle
- Definition: 4.7: Chord and Secant
- Definition: Altitude of a Triangle
- Definition: Collinear Points, Segments, Rays
- Definition: Diagonal
- Definition: Exterior, Interior, Alternate and Corresponding Angles
- Definition: Parallelogram - Defining Property III
- Definition: Point of Division, Point of External Division
- Definition: Sum of Angles
- Definition: Supplemental Angles
- Explanation: How a line is different from a solid and a surface? (related to Definition: 1.02: Line, Curve)
- Explanation: How a point is different from a solid, a surface and a line? (related to Definition: 1.01: Point)
- Explanation: Why did Euclid postulate the axiom of straight line determined by two points? (related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
- Part: Historical Development of Geometry
- Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
- Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
- Proof: By Euclid (related to Corollary: Angles of Right Triangle)
- Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
- Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
- Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
- Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
- Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
- Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
- Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
- Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
- Proof: By Euclid (related to Corollary: Similar Triangles)
- Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
- Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
- Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
- Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
- Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
- Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proposition: 1.01: Constructing an Equilateral Triangle
- Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment
- Proposition: 1.03: Cutting a Segment at a Given Size
- Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
- Proposition: 1.05: Isosceles Triangles I
- Proposition: 1.06: Isosceles Triagles II
- Proposition: 1.07: Uniqueness of Triangles
- Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
- Proposition: 1.09: Bisecting an Angle
- Proposition: 1.10: Bisecting a Segment
- Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line
- Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
- Proposition: 1.13: Angles at Intersections of Straight Lines
- Proposition: 1.14: Combining Rays to Straight Lines
- Proposition: 1.15: Opposite Angles on Intersecting Straight Lines
- Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
- Proposition: 1.17: The Sum of Two Angles of a Triangle
- Proposition: 1.18: Angles and Sides in a Triangle I
- Proposition: 1.19: Angles and Sides in a Triangle II
- Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
- Proposition: 1.21: Triangles within Triangles
- Proposition: 1.22: Construction of Triangles From Arbitrary Segments
- Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
- Proposition: 1.24: Angles and Sides in a Triangle III
- Proposition: 1.25: Angles and Sides in a Triangle IV
- Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
- Proposition: 1.27: Parallel Lines I
- Proposition: 1.28: Parallel Lines II
- Proposition: 1.29: Parallel Lines III
- Proposition: 1.30: Transitivity of Parallel Lines
- Proposition: 1.31: Constructing a Parallel Line from a Line and a Point
- Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle
- Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
- Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
- Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels
- Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels
- Proposition: 1.37: Triangles of Equal Area I
- Proposition: 1.38: Triangles of Equal Area II
- Proposition: 1.39: Triangles of Equal Area III
- Proposition: 1.40: Triangles of Equal Area IV
- Proposition: 1.41: Parallelograms and Triagles
- Proposition: 1.42: Construction of Parallelograms I
- Proposition: 1.43: Complementary Segments of Parallelograms
- Proposition: 1.44: Construction of Parallelograms II
- Proposition: 1.45: Construction of Parallelograms III
- Proposition: 1.46: Construction of a Square on a Given Segment
- Proposition: 1.47: Pythagorean Theorem
- Proposition: 1.48: The Converse of the Pythagorean Theorem
- Proposition: 2.01: Summing Areas or Rectangles
- Proposition: 2.04: Square of Sum
- Proposition: 2.05: Rectangle is Difference of Two Squares
- Proposition: 2.06: Square of Sum with One Halved Summand
- Proposition: 2.07: Sum of Squares
- Proposition: 2.08: Square of Sum with One Doubled Summand
- Proposition: 2.09: Sum of Squares of Sum and Difference
- Proposition: 2.10: Sum of Squares (Half)
- Proposition: 2.11: Constructing the Golden Ratio of a Segment
- Proposition: 2.14: Constructing a Square from a Rectilinear Figure
- Proposition: 3.01: Finding the Center of a given Circle
- Proposition: 3.02: Chord Lies Inside its Circle
- Proposition: 7.03: Greatest Common Divisor of Three Numbers
- Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines
- Section: Book 02: Fundamentals of Geometric Algebra
- Section: Book 03: Fundamentals of Plane Geometry Involving Circles
- Section: Book 06: Similar Figures
- Subsection: Definitions from Book 4
- @Casey (135)
- Axiom: 1.1: Straight Line Determined by Two Distinct Points
- Axiom: 1.2: Segment Extension
- Axiom: 1.3: Circle Determined by its Center and its Radius
- Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle)
- Corollary: A Criterion for Isosceles Triangles (related to Proposition: 1.06: Isosceles Triagles II)
- Corollary: Angles and Sides in a Triangle V (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Corollary: Angles of Right Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Angles of a Right And Isosceles Triangle (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: Diagonals of a Rectangle (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Diagonals of a Rhombus (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Equivalent Statements Regarding Parallel Lines (related to Proposition: 1.29: Parallel Lines III)
- Corollary: Every Equilateral Triangle Is Equiangular. (related to Proposition: 1.05: Isosceles Triangles I)
- Corollary: Existence of Parallel Straight Lines (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Corollary: Parallelogram - Defining Property I (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Parallelogram - Defining Property II (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rectangle as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Rhombus as a Special Case of a Parallelogram (related to Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms)
- Corollary: Similar Triangles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Square as a Special Case of a Rhombus (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Corollary: Sum of Two Supplemental Angles Equals Two Right Angles (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Corollary: The supplemental angle of a right angle is another right angle. (related to Definition: 1.10: Right Angle, Perpendicular Straight Lines)
- Corollary: Triangulation of Quadrilateral and Sum of Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Corollary: Triangulation of an N-gon and Sum of Interior Angles (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Definition: 1.02: Line, Curve
- Definition: 1.03: Intersections of Lines
- Definition: 1.05: Surface
- Definition: 1.06: Intersections of Surfaces
- Definition: 1.09: Angle, Rectilinear, Vertex, Legs
- Definition: 1.10: Right Angle, Perpendicular Straight Lines
- Definition: 1.11: Obtuse Angle
- Definition: 1.15: Circle, Circumference, Radius
- Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure
- Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle
- Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium
- Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides
- Definition: 3.03: Circles Touching One Another
- Definition: 4.7: Chord and Secant
- Definition: Altitude of a Triangle
- Definition: Collinear Points, Segments, Rays
- Definition: Concentric Circles
- Definition: Diagonal
- Definition: Exterior, Interior, Alternate and Corresponding Angles
- Definition: Parallelogram - Defining Property III
- Definition: Point of Division, Point of External Division
- Definition: Sum of Angles
- Definition: Supplemental Angles
- Explanation: How a line is different from a solid and a surface? (related to Definition: 1.02: Line, Curve)
- Explanation: How a point is different from a solid, a surface and a line? (related to Definition: 1.01: Point)
- Explanation: Why did Euclid postulate the axiom of straight line determined by two points? (related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
- Proof: By Euclid (related to Corollary: A Criterion for Isosceles Triangles)
- Proof: By Euclid (related to Corollary: Angles and Sides in a Triangle V)
- Proof: By Euclid (related to Corollary: Angles of Right Triangle)
- Proof: By Euclid (related to Corollary: Angles of a Right And Isosceles Triangle)
- Proof: By Euclid (related to Corollary: Bisectors of Two Supplemental Angles Are Right Angle To Each Other)
- Proof: By Euclid (related to Corollary: Diagonals of a Rectangle)
- Proof: By Euclid (related to Corollary: Diagonals of a Rhombus)
- Proof: By Euclid (related to Corollary: Equivalent Statements Regarding Parallel Lines)
- Proof: By Euclid (related to Corollary: Every Equilateral Triangle Is Equiangular.)
- Proof: By Euclid (related to Corollary: Existence of Parallel Straight Lines)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property I)
- Proof: By Euclid (related to Corollary: Parallelogram - Defining Property II)
- Proof: By Euclid (related to Corollary: Rectangle as a Special Case of a Parallelogram)
- Proof: By Euclid (related to Corollary: Similar Triangles)
- Proof: By Euclid (related to Corollary: Square as a Special Case of a Rhombus)
- Proof: By Euclid (related to Corollary: Sum of Two Supplemental Angles Equals Two Right Angles)
- Proof: By Euclid (related to Corollary: Triangulation of Quadrilateral and Sum of Angles)
- Proof: By Euclid (related to Corollary: Triangulation of an N-gon and Sum of Interior Angles)
- Proof: By Euclid (related to Corollary: 3.01: Bisected Chord of a Circle Passes the Center)
- Proof: By Euclid (related to Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proposition: 1.01: Constructing an Equilateral Triangle
- Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment
- Proposition: 1.03: Cutting a Segment at a Given Size
- Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
- Proposition: 1.05: Isosceles Triangles I
- Proposition: 1.06: Isosceles Triagles II
- Proposition: 1.07: Uniqueness of Triangles
- Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
- Proposition: 1.09: Bisecting an Angle
- Proposition: 1.10: Bisecting a Segment
- Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line
- Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
- Proposition: 1.13: Angles at Intersections of Straight Lines
- Proposition: 1.14: Combining Rays to Straight Lines
- Proposition: 1.15: Opposite Angles on Intersecting Straight Lines
- Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
- Proposition: 1.17: The Sum of Two Angles of a Triangle
- Proposition: 1.18: Angles and Sides in a Triangle I
- Proposition: 1.19: Angles and Sides in a Triangle II
- Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
- Proposition: 1.21: Triangles within Triangles
- Proposition: 1.22: Construction of Triangles From Arbitrary Segments
- Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
- Proposition: 1.24: Angles and Sides in a Triangle III
- Proposition: 1.25: Angles and Sides in a Triangle IV
- Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
- Proposition: 1.27: Parallel Lines I
- Proposition: 1.28: Parallel Lines II
- Proposition: 1.29: Parallel Lines III
- Proposition: 1.30: Transitivity of Parallel Lines
- Proposition: 1.31: Constructing a Parallel Line from a Line and a Point
- Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle
- Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
- Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
- Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels
- Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels
- Proposition: 1.37: Triangles of Equal Area I
- Proposition: 1.38: Triangles of Equal Area II
- Proposition: 1.39: Triangles of Equal Area III
- Proposition: 1.40: Triangles of Equal Area IV
- Proposition: 1.41: Parallelograms and Triagles
- Proposition: 1.42: Construction of Parallelograms I
- Proposition: 1.43: Complementary Segments of Parallelograms
- Proposition: 1.44: Construction of Parallelograms II
- Proposition: 1.45: Construction of Parallelograms III
- Proposition: 1.46: Construction of a Square on a Given Segment
- Proposition: 1.47: Pythagorean Theorem
- Proposition: 1.48: The Converse of the Pythagorean Theorem
- Proposition: 2.01: Summing Areas or Rectangles
- Proposition: 2.04: Square of Sum
- Proposition: 2.05: Rectangle is Difference of Two Squares
- Proposition: 2.06: Square of Sum with One Halved Summand
- Proposition: 2.07: Sum of Squares
- Proposition: 2.08: Square of Sum with One Doubled Summand
- Proposition: 2.09: Sum of Squares of Sum and Difference
- Proposition: 2.10: Sum of Squares (Half)
- Proposition: 2.11: Constructing the Golden Ratio of a Segment
- Proposition: 2.14: Constructing a Square from a Rectilinear Figure
- Proposition: 3.01: Finding the Center of a given Circle
- Proposition: 3.02: Chord Lies Inside its Circle
- Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines
- Section: Book 02: Fundamentals of Geometric Algebra
- Section: Book 03: Fundamentals of Plane Geometry Involving Circles
- Section: Book 06: Similar Figures
- @Catherine-Felgate (1)
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- @E-F-Robertson (3035)
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- Axiom: 1.1: Straight Line Determined by Two Distinct Points
- Axiom: 1.2: Segment Extension
- Axiom: 1.3: Circle Determined by its Center and its Radius
- Axiom: 1.4: Equality of all Right Angles
- Axiom: 1.5: Parallel Postulate
- Chapter: Euclid's “Elements”
- Corollary: 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
- Corollary: 3.01: Bisected Chord of a Circle Passes the Center (related to Proposition: 3.01: Finding the Center of a given Circle)
- Corollary: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Corollary: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Corollary: 5.07: Ratios of Equal Magnitudes (related to Proposition: 5.07: Ratios of Equal Magnitudes)
- Corollary: 5.19: Proportional Magnitudes have Proportional Remainders (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
- Corollary: 6.08: Geometric Mean Theorem (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
- Corollary: 6.19: Ratio of Areas of Similar Triangles (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
- Corollary: 6.20: Similar Polygons are Composed of Similar Triangles (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
- Corollary: 7.02: Any Divisor Dividing Two Numbers Divides Their Greatest Common Divisor (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
- Corollary: 8.02: Construction of Geometric Progression in Lowest Terms (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
- Corollary: 9.11: Elements of Geometric Progression from One which Divide Later Elements (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Corollary: Cor. 10.003: Greatest Common Measure of Commensurable Magnitudes (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Corollary: Cor. 10.004: Greatest Common Measure of Three Commensurable Magnitudes (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Corollary: Cor. 10.006: Magnitudes with Rational Ratio are Commensurable (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Corollary: Cor. 10.009: Commensurability of Squares (related to Proposition: Prop. 10.009: Commensurability of Squares)
- Corollary: Cor. 10.023: Segment Commensurable with Medial Area is Medial (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
- Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
- Corollary: Cor. 10.114: Rectangles With Irrational Sides Can Have Rational Areas (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)
- Corollary: Cor. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
- Corollary: Cor. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
- Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
- Corollary: Cor. 13.16: Construction of Regular Icosahedron within Given Sphere (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
- Corollary: Cor. 13.17: Construction of Regular Dodecahedron within Given Sphere (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
- Definition: 1.02: Line, Curve
- Definition: 1.03: Intersections of Lines
- Definition: 1.05: Surface
- Definition: 1.06: Intersections of Surfaces
- Definition: 1.08: Plane Angle
- Definition: 1.09: Angle, Rectilinear, Vertex, Legs
- Definition: 1.10: Right Angle, Perpendicular Straight Lines
- Definition: 1.11: Obtuse Angle
- Definition: 1.12: Acute Angle
- Definition: 1.13: Boundary
- Definition: 1.14: Plane Figure
- Definition: 1.15: Circle, Circumference, Radius
- Definition: 1.16: Center of the Circle
- Definition: 1.17: Diameter of the Circle
- Definition: 1.18: Semicircle
- Definition: 1.19: Rectilinear Figure, Sides, n-Sided Figure
- Definition: 1.20: Equilateral Triangle, Isosceles Triangle, Scalene Triangle
- Definition: 1.21: Right Triangle, Obtuse Triangle, Acute Triangle
- Definition: 1.22: Square, Rectangle, Rhombus, Rhomboid, Trapezium
- Definition: 1.23: Parallel Straight Lines
- Definition: 10.02: Magnitudes Commensurable and Incommensurable in Square
- Definition: 2.1: Area of Rectangle, Rectangle Contained by Adjacent Sides
- Definition: 2.2: Gnomon
- Definition: 3.01: Congruent Circles
- Definition: 3.02: Tangent to the Circle, Straight Line Touching The Circle
- Definition: 3.03: Circles Touching One Another
- Definition: 3.04: Chords Equally Far From the Center of a Circle
- Definition: 3.05: Chords Being Further from the Center of a Circle
- Definition: 3.06: Segment of a Circle, Arc
- Definition: 3.07: Angle of a Segment
- Definition: 3.08: Angle in the Segment (Inscribed Angle)
- Definition: 3.09: Angle Standing Upon An Arc
- Definition: 3.10: Circular Sector, Central Angle
- Definition: 3.11: Similar Circular Segments
- Definition: 4.1: Rectilinear Figure Inscribed in Another Rectilinear Figure
- Definition: 4.2: Rectilinear Figure Circumscribed about Another Rectilinear Figure
- Definition: 4.3: Inscribing Rectilinear Figures in Circles
- Definition: 4.4: Circumscribing Rectilinear Figures about Circles
- Definition: 4.5: Inscribing Circles in Rectilinear Figures
- Definition: 4.6: Circumscribing Circles about Rectilinear Figures
- Definition: 4.7: Chord and Secant
- Definition: 5.01: Magnitude is Aliquot Part
- Definition: 5.02: Multiple of a Real Number
- Definition: 5.03: Ratio
- Definition: 5.04: Having a Ratio
- Definition: 5.05: Having the Same Ratio
- Definition: 5.06: Proportional Magnitudes
- Definition: 5.07: Having a Greater Ratio
- Definition: 5.08: Proportion in Three Terms
- Definition: 5.09: Squared Ratio
- Definition: 5.10: Cubed Ratio
- Definition: 5.12: Alternate Ratio
- Definition: 5.13: Inverse Ratio
- Definition: 5.14: Composition of a Ratio
- Definition: 5.15: Separation of a Ratio
- Definition: 5.16: Conversion of a Ratio
- Definition: 5.17: Ratio ex Aequali
- Definition: 5.18: Perturbed Proportion
- Definition: 6.01: Similar Rectilinear Figures
- Definition: 6.02: Cut in Extreme and Mean Ratio
- Definition: 6.03: Height of a Figure
- Definition: 7.01: Unit
- Definition: 7.02: Number
- Definition: 7.03: Proper Divisor
- Definition: 7.04: Aliquant Part, a Number Being Not a Divisor of Another Number
- Definition: 7.05: Multiple, Number Multiplying another Number
- Definition: 7.06: Even Number
- Definition: 7.07: Odd Number
- Definition: 7.08: Even-Times-Even Number
- Definition: 7.09: Even-Times-Odd Number
- Definition: 7.10: Odd-Times-Odd Number
- Definition: 7.11: Prime Number
- Definition: 7.12: Co-prime (Relatively Prime) Numbers
- Definition: 7.13: Composite Number
- Definition: 7.14: Not Co-prime Numbers
- Definition: 7.15: Multiplication of Numbers
- Definition: 7.16: Rectangular Number, Plane Number
- Definition: 7.17: Cuboidal Number, Solid Number
- Definition: 7.18: Square Number
- Definition: 7.19: Cubic Number, Cube Number
- Definition: 7.20: Proportional Numbers
- Definition: 7.21: Similar Rectangles and Similar Cuboids, Similar Plane and Solid Numbers
- Definition: 7.22: Perfect Number
- Definition: Def. 10.01: Magnitudes Commensurable and Incommensurable in Length
- Definition: Def. 10.03: Rational and Irrational Magnitudes
- Definition: Def. 10.04: Rational and Irrational Magnitudes in Square
- Definition: Def. 10.05: First Binomial
- Definition: Def. 10.06: Second Binomial
- Definition: Def. 10.07: Third Binomial
- Definition: Def. 10.08: Fourth Binomial
- Definition: Def. 10.09: Fifth Binomial
- Definition: Def. 10.10: Sixth Binomial
- Definition: Def. 10.11: First Apotome
- Definition: Def. 10.12: Second Apotome
- Definition: Def. 10.13: Third Apotome
- Definition: Def. 10.14: Fourth Apotome
- Definition: Def. 10.15: Fifth Apotome
- Definition: Def. 10.16: Sixth Apotome
- Definition: Def. 11.01: Solid Figures, Three-Dimensional Polyhedra
- Definition: Def. 11.02: Surface of a Solid Figure
- Definition: Def. 11.03: Straight Line at Right Angles To a Plane
- Definition: Def. 11.04: Plane at Right Angles to a Plane
- Definition: Def. 11.05: Inclination of a Straight Line to a Plane
- Definition: Def. 11.06: Inclination of a Plane to a Plane
- Definition: Def. 11.07: Similarly Inclined Planes
- Definition: Def. 11.08: Parallel Planes
- Definition: Def. 11.09: Similar Solid Figures
- Definition: Def. 11.10: Equal Solid Figures
- Definition: Def. 11.11: Solid Angle
- Definition: Def. 11.12: Pyramid, Tetrahedron
- Definition: Def. 11.13: Prism, Parallelepiped
- Definition: Def. 11.14: Sphere
- Definition: Def. 11.15: Axis of a Sphere
- Definition: Def. 11.16: Center of a Sphere
- Definition: Def. 11.17: Diameter of a Sphere
- Definition: Def. 11.18: Cone
- Definition: Def. 11.19: Axis of a Cone
- Definition: Def. 11.20: Base of a Cone
- Definition: Def. 11.21: Cylinder
- Definition: Def. 11.22: Axis of a Cylinder
- Definition: Def. 11.23: Bases of a Cylinder
- Definition: Def. 11.24: Similar Cones, Similar Cylinders
- Definition: Def. 11.25: Cube
- Definition: Def. 11.26: Octahedron
- Definition: Def. 11.27: Icosahedron
- Definition: Def. 11.28: Dodecahedron
- Explanation: 1.1: Equality is an Equivalence Relation (related to Subsection: Common Notions (all Books))
- Explanation: 1.2: Adding Equations Preserves Equality (related to Subsection: Common Notions (all Books))
- Explanation: 1.3: Subtracting Equations Preserves Equality (related to Subsection: Common Notions (all Books))
- Explanation: 1.5: Comparing the Size of Sets and Their Subsets (related to Subsection: Common Notions (all Books))
- Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Lemma: Lem. 10.021: Medial is Irrational
- Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square
- Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square
- Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II
- Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational
- Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas
- Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them
- Lemma: Lem. 10.13: Finding Pythagorean Magnitudes
- Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares
- Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters
- Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere
- Lemma: Lem. 13.18: Angle of the Pentagon
- Proof: (Euclid) (related to Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line)
- Proof: (Euclid) (related to Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line)
- Proof: (Euclid) (related to Proposition: 1.18: Angles and Sides in a Triangle I)
- Proof: (Euclid) (related to Proposition: 1.19: Angles and Sides in a Triangle II)
- Proof: (Euclid) (related to Proposition: 1.24: Angles and Sides in a Triangle III)
- Proof: (Euclid) (related to Proposition: 1.25: Angles and Sides in a Triangle IV)
- Proof: (Euclid) (related to Proposition: 1.31: Constructing a Parallel Line from a Line and a Point)
- Proof: (Euclid) (related to Proposition: 1.43: Complementary Segments of Parallelograms)
- Proof: By Euclid (related to Proposition: 1.01: Constructing an Equilateral Triangle)
- Proof: By Euclid (related to Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment)
- Proof: By Euclid (related to Proposition: 1.03: Cutting a Segment at a Given Size)
- Proof: By Euclid (related to Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle)
- Proof: By Euclid (related to Proposition: 1.05: Isosceles Triangles I)
- Proof: By Euclid (related to Proposition: 1.06: Isosceles Triagles II)
- Proof: By Euclid (related to Proposition: 1.07: Uniqueness of Triangles)
- Proof: By Euclid (related to Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles)
- Proof: By Euclid (related to Proposition: 1.09: Bisecting an Angle)
- Proof: By Euclid (related to Proposition: 1.10: Bisecting a Segment)
- Proof: By Euclid (related to Proposition: 1.13: Angles at Intersections of Straight Lines)
- Proof: By Euclid (related to Proposition: 1.14: Combining Rays to Straight Lines)
- Proof: By Euclid (related to Proposition: 1.15: Opposite Angles on Intersecting Straight Lines)
- Proof: By Euclid (related to Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles)
- Proof: By Euclid (related to Proposition: 1.17: The Sum of Two Angles of a Triangle)
- Proof: By Euclid (related to Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality))
- Proof: By Euclid (related to Proposition: 1.21: Triangles within Triangles)
- Proof: By Euclid (related to Proposition: 1.22: Construction of Triangles From Arbitrary Segments)
- Proof: By Euclid (related to Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle)
- Proof: By Euclid (related to Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles)
- Proof: By Euclid (related to Proposition: 1.27: Parallel Lines I)
- Proof: By Euclid (related to Proposition: 1.28: Parallel Lines II)
- Proof: By Euclid (related to Proposition: 1.29: Parallel Lines III)
- Proof: By Euclid (related to Proposition: 1.30: Transitivity of Parallel Lines)
- Proof: By Euclid (related to Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle)
- Proof: By Euclid (related to Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram)
- Proof: By Euclid (related to Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels)
- Proof: By Euclid (related to Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels)
- Proof: By Euclid (related to Proposition: 1.37: Triangles of Equal Area I)
- Proof: By Euclid (related to Proposition: 1.38: Triangles of Equal Area II)
- Proof: By Euclid (related to Proposition: 1.39: Triangles of Equal Area III)
- Proof: By Euclid (related to Proposition: 1.40: Triangles of Equal Area IV)
- Proof: By Euclid (related to Proposition: 1.41: Parallelograms and Triagles)
- Proof: By Euclid (related to Proposition: 1.42: Construction of Parallelograms I)
- Proof: By Euclid (related to Proposition: 1.44: Construction of Parallelograms II)
- Proof: By Euclid (related to Proposition: 1.46: Construction of a Square on a Given Segment)
- Proof: By Euclid (related to Proposition: 1.47: Pythagorean Theorem)
- Proof: By Euclid (related to Proposition: 1.48: The Converse of the Pythagorean Theorem)
- Proof: By Euclid (related to Proposition: 2.01: Summing Areas or Rectangles)
- Proof: By Euclid (related to Proposition: 2.02: Square is Sum of Two Rectangles)
- Proof: By Euclid (related to Proposition: 2.03: Rectangle is Sum of Square and Rectangle)
- Proof: By Euclid (related to Proposition: 2.04: Square of Sum)
- Proof: By Euclid (related to Proposition: 2.05: Rectangle is Difference of Two Squares)
- Proof: By Euclid (related to Proposition: 2.06: Square of Sum with One Halved Summand)
- Proof: By Euclid (related to Proposition: 2.07: Sum of Squares)
- Proof: By Euclid (related to Proposition: 2.08: Square of Sum with One Doubled Summand)
- Proof: By Euclid (related to Proposition: 2.09: Sum of Squares of Sum and Difference)
- Proof: By Euclid (related to Proposition: 2.10: Sum of Squares (Half))
- Proof: By Euclid (related to Proposition: 2.11: Constructing the Golden Ratio of a Segment)
- Proof: By Euclid (related to Proposition: 2.12: Law of Cosines (for Obtuse Angles))
- Proof: By Euclid (related to Proposition: 2.13: Law of Cosines (for Acute Angles))
- Proof: By Euclid (related to Proposition: 2.14: Constructing a Square from a Rectilinear Figure)
- Proof: By Euclid (related to Proposition: 3.01: Finding the Center of a given Circle)
- Proof: By Euclid (related to Proposition: 3.02: Chord Lies Inside its Circle)
- Proof: By Euclid (related to Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector)
- Proof: By Euclid (related to Proposition: 3.04: Chords do not Bisect Each Other)
- Proof: By Euclid (related to Proposition: 3.05: Intersecting Circles have Different Centers)
- Proof: By Euclid (related to Proposition: 3.06: Touching Circles have Different Centers)
- Proof: By Euclid (related to Proposition: 3.07: Relative Lengths of Lines Inside Circle)
- Proof: By Euclid (related to Proposition: 3.08: Relative Lengths of Lines Outside Circle)
- Proof: By Euclid (related to Proposition: 3.09: Condition for Point to be Center of Circle)
- Proof: By Euclid (related to Proposition: 3.10: Two Circles have at most Two Points of Intersection)
- Proof: By Euclid (related to Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally)
- Proof: By Euclid (related to Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally)
- Proof: By Euclid (related to Proposition: 3.13: Circles Touch at One Point at Most)
- Proof: By Euclid (related to Proposition: 3.14: Equal Chords in Circle)
- Proof: By Euclid (related to Proposition: 3.15: Relative Lengths of Chords of Circles)
- Proof: By Euclid (related to Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle)
- Proof: By Euclid (related to Proposition: 3.17: Construction of Tangent from Point to Circle)
- Proof: By Euclid (related to Proposition: 3.18: Radius at Right Angle to Tangent)
- Proof: By Euclid (related to Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center)
- Proof: By Euclid (related to Proposition: 3.20: Inscribed Angle Theorem)
- Proof: By Euclid (related to Proposition: 3.21: Angles in Same Segment of Circle are Equal)
- Proof: By Euclid (related to Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral)
- Proof: By Euclid (related to Proposition: 3.23: Segment on Given Base Unique)
- Proof: By Euclid (related to Proposition: 3.24: Similar Segments on Equal Bases are Equal)
- Proof: By Euclid (related to Proposition: 3.25: Construction of Circle from Segment)
- Proof: By Euclid (related to Proposition: 3.26: Equal Angles and Arcs in Equal Circles)
- Proof: By Euclid (related to Proposition: 3.27: Angles on Equal Arcs are Equal)
- Proof: By Euclid (related to Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles)
- Proof: By Euclid (related to Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines)
- Proof: By Euclid (related to Proposition: 3.30: Bisection of Arc)
- Proof: By Euclid (related to Proposition: 3.31: Relative Sizes of Angles in Segments)
- Proof: By Euclid (related to Proposition: 3.32: Angles made by Chord with Tangent)
- Proof: By Euclid (related to Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle)
- Proof: By Euclid (related to Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle)
- Proof: By Euclid (related to Proposition: 3.35: Intersecting Chord Theorem)
- Proof: By Euclid (related to Proposition: 3.36: Tangent Secant Theorem)
- Proof: By Euclid (related to Proposition: 3.37: Converse of Tangent Secant Theorem)
- Proof: By Euclid (related to Proposition: 4.01: Fitting Chord Into Circle)
- Proof: By Euclid (related to Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles)
- Proof: By Euclid (related to Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles)
- Proof: By Euclid (related to Proposition: 4.04: Inscribing Circle in Triangle)
- Proof: By Euclid (related to Proposition: 4.05: Circumscribing Circle about Triangle)
- Proof: By Euclid (related to Proposition: 4.06: Inscribing Square in Circle)
- Proof: By Euclid (related to Proposition: 4.07: Circumscribing Square about Circle)
- Proof: By Euclid (related to Proposition: 4.08: Inscribing Circle in Square)
- Proof: By Euclid (related to Proposition: 4.09: Circumscribing Circle about Square)
- Proof: By Euclid (related to Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex)
- Proof: By Euclid (related to Proposition: 4.11: Inscribing Regular Pentagon in Circle)
- Proof: By Euclid (related to Proposition: 4.12: Circumscribing Regular Pentagon about Circle)
- Proof: By Euclid (related to Proposition: 4.13: Inscribing Circle in Regular Pentagon)
- Proof: By Euclid (related to Proposition: 4.14: Circumscribing Circle about Regular Pentagon)
- Proof: By Euclid (related to Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle)
- Proof: By Euclid (related to Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle)
- Proof: By Euclid (related to Proposition: 5.01: Multiplication of Numbers is Left Distributive over Addition)
- Proof: By Euclid (related to Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition)
- Proof: By Euclid (related to Proposition: 5.03: Multiplication of Numbers is Associative)
- Proof: By Euclid (related to Proposition: 5.04: Multiples of Terms in Equal Ratios)
- Proof: By Euclid (related to Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction)
- Proof: By Euclid (related to Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction)
- Proof: By Euclid (related to Proposition: 5.07: Ratios of Equal Magnitudes)
- Proof: By Euclid (related to Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes)
- Proof: By Euclid (related to Proposition: 5.09: Magnitudes with Same Ratios are Equal)
- Proof: By Euclid (related to Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios)
- Proof: By Euclid (related to Proposition: 5.11: Equality of Ratios is Transitive)
- Proof: By Euclid (related to Proposition: 5.12: Sum of Components of Equal Ratios)
- Proof: By Euclid (related to Proposition: 5.13: Relative Sizes of Proportional Magnitudes)
- Proof: By Euclid (related to Proposition: 5.14: Relative Sizes of Components of Ratios)
- Proof: By Euclid (related to Proposition: 5.15: Ratio Equals its Multiples)
- Proof: By Euclid (related to Proposition: 5.16: Proportional Magnitudes are Proportional Alternately)
- Proof: By Euclid (related to Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated)
- Proof: By Euclid (related to Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded)
- Proof: By Euclid (related to Proposition: 5.19: Proportional Magnitudes have Proportional Remainders)
- Proof: By Euclid (related to Proposition: 5.20: Relative Sizes of Successive Ratios)
- Proof: By Euclid (related to Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion)
- Proof: By Euclid (related to Proposition: 5.22: Equality of Ratios Ex Aequali)
- Proof: By Euclid (related to Proposition: 5.23: Equality of Ratios in Perturbed Proportion)
- Proof: By Euclid (related to Proposition: 5.24: Sum of Antecedents of Proportion)
- Proof: By Euclid (related to Proposition: 5.25: Sum of Antecedent and Consequent of Proportion)
- Proof: By Euclid (related to Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base)
- Proof: By Euclid (related to Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally)
- Proof: By Euclid (related to Proposition: 6.03: Angle Bisector Theorem)
- Proof: By Euclid (related to Proposition: 6.04: Equiangular Triangles are Similar)
- Proof: By Euclid (related to Proposition: 6.05: Triangles with Proportional Sides are Similar)
- Proof: By Euclid (related to Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar)
- Proof: By Euclid (related to Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar)
- Proof: By Euclid (related to Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.09: Construction of Part of Line)
- Proof: By Euclid (related to Proposition: 6.10: Construction of Similarly Cut Straight Line)
- Proof: By Euclid (related to Proposition: 6.11: Construction of Segment in Squared Ratio)
- Proof: By Euclid (related to Proposition: 6.12: Construction of Fourth Proportional Straight Line)
- Proof: By Euclid (related to Proposition: 6.13: Construction of Mean Proportional)
- Proof: By Euclid (related to Proposition: 6.14: Characterization of Congruent Parallelograms)
- Proof: By Euclid (related to Proposition: 6.15: Characterization of Congruent Triangles)
- Proof: By Euclid (related to Proposition: 6.16: Rectangles Contained by Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.18: Construction of Similar Polygon)
- Proof: By Euclid (related to Proposition: 6.19: Ratio of Areas of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.20: Similar Polygons are Composed of Similar Triangles)
- Proof: By Euclid (related to Proposition: 6.21: Similarity of Polygons is Transitive)
- Proof: By Euclid (related to Proposition: 6.22: Similar Figures on Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms)
- Proof: By Euclid (related to Proposition: 6.24: Parallelograms About Diameter are Similar)
- Proof: By Euclid (related to Proposition: 6.25: Construction of Figure Similar to One and Equal to Another)
- Proof: By Euclid (related to Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter)
- Proof: By Euclid (related to Proposition: 6.27: Similar Parallelogram on Half a Straight Line)
- Proof: By Euclid (related to Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram)
- Proof: By Euclid (related to Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram)
- Proof: By Euclid (related to Proposition: 6.30: Construction of the Inverse Golden Section)
- Proof: By Euclid (related to Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle)
- Proof: By Euclid (related to Proposition: 6.32: Triangles with Two Sides Parallel and Equal)
- Proof: By Euclid (related to Proposition: 6.33: Angles in Circles have Same Ratio as Arcs)
- Proof: By Euclid (related to Proposition: 7.01: Sufficient Condition for Coprimality)
- Proof: By Euclid (related to Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm)
- Proof: By Euclid (related to Proposition: 7.03: Greatest Common Divisor of Three Numbers)
- Proof: By Euclid (related to Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers)
- Proof: By Euclid (related to Proposition: 7.05: Divisors Obey Distributive Law (Sum))
- Proof: By Euclid (related to Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum))
- Proof: By Euclid (related to Proposition: 7.07: Divisors Obey Distributive Law (Difference))
- Proof: By Euclid (related to Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference))
- Proof: By Euclid (related to Proposition: 7.09: Alternate Ratios of Equal Fractions)
- Proof: By Euclid (related to Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions)
- Proof: By Euclid (related to Proposition: 7.11: Proportional Numbers have Proportional Differences)
- Proof: By Euclid (related to Proposition: 7.12: Ratios of Numbers is Distributive over Addition)
- Proof: By Euclid (related to Proposition: 7.13: Proportional Numbers are Proportional Alternately)
- Proof: By Euclid (related to Proposition: 7.14: Proportion of Numbers is Transitive)
- Proof: By Euclid (related to Proposition: 7.15: Alternate Ratios of Multiples)
- Proof: By Euclid (related to Proposition: 7.16: Natural Number Multiplication is Commutative)
- Proof: By Euclid (related to Proposition: 7.17: Multiples of Ratios of Numbers)
- Proof: By Euclid (related to Proposition: 7.18: Ratios of Multiples of Numbers)
- Proof: By Euclid (related to Proposition: 7.19: Relation of Ratios to Products)
- Proof: By Euclid (related to Proposition: 7.20: Ratios of Fractions in Lowest Terms)
- Proof: By Euclid (related to Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms)
- Proof: By Euclid (related to Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime)
- Proof: By Euclid (related to Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other)
- Proof: By Euclid (related to Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole)
- Proof: By Euclid (related to Proposition: 7.25: Square of Co-prime Number is Co-prime)
- Proof: By Euclid (related to Proposition: 7.26: Product of Co-prime Pairs is Co-prime)
- Proof: By Euclid (related to Proposition: 7.27: Powers of Co-prime Numbers are Co-prime)
- Proof: By Euclid (related to Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both)
- Proof: By Euclid (related to Proposition: 7.29: Prime not Divisor implies Co-prime)
- Proof: By Euclid (related to Proposition: 7.30: Euclidean Lemma)
- Proof: By Euclid (related to Proposition: 7.31: Existence of Prime Divisors)
- Proof: By Euclid (related to Proposition: 7.32: Natural Number is Prime or has Prime Factor)
- Proof: By Euclid (related to Proposition: 7.33: Least Ratio of Numbers)
- Proof: By Euclid (related to Proposition: 7.34: Existence of Least Common Multiple)
- Proof: By Euclid (related to Proposition: 7.35: Least Common Multiple Divides Common Multiple)
- Proof: By Euclid (related to Proposition: 7.36: Least Common Multiple of Three Numbers)
- Proof: By Euclid (related to Proposition: 7.37: Integer Divided by Divisor is Integer)
- Proof: By Euclid (related to Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer)
- Proof: By Euclid (related to Proposition: 7.39: Least Number with Three Given Fractions)
- Proof: By Euclid (related to Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.02: Construction of Geometric Progression in Lowest Terms)
- Proof: By Euclid (related to Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes)
- Proof: By Euclid (related to Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios)
- Proof: By Euclid (related to Proposition: 8.05: Ratio of Products of Sides of Plane Numbers)
- Proof: By Euclid (related to Proposition: 8.06: First Element of Geometric Progression not dividing Second)
- Proof: By Euclid (related to Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second)
- Proof: By Euclid (related to Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements)
- Proof: By Euclid (related to Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers)
- Proof: By Euclid (related to Proposition: Prop. 8.10: Product of Geometric Progressions from One)
- Proof: By Euclid (related to Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional)
- Proof: By Euclid (related to Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals)
- Proof: By Euclid (related to Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression)
- Proof: By Euclid (related to Proposition: Prop. 8.14: Number divides Number iff Square divides Square)
- Proof: By Euclid (related to Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square)
- Proof: By Euclid (related to Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional)
- Proof: By Euclid (related to Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals)
- Proof: By Euclid (related to Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane)
- Proof: By Euclid (related to Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid)
- Proof: By Euclid (related to Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square)
- Proof: By Euclid (related to Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square)
- Proof: By Euclid (related to Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares)
- Proof: By Euclid (related to Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes)
- Proof: By Euclid (related to Proposition: 9.35: Sum of Geometric Progression)
- Proof: By Euclid (related to Proposition: 9.36: Theorem of Even Perfect Numbers (First Part))
- Proof: By Euclid (related to Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square)
- Proof: By Euclid (related to Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.03: Square of Cube Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.06: Number Squared making Cube is itself Cube)
- Proof: By Euclid (related to Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number)
- Proof: By Euclid (related to Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number)
- Proof: By Euclid (related to Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements)
- Proof: By Euclid (related to Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime)
- Proof: By Euclid (related to Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime)
- Proof: By Euclid (related to Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element)
- Proof: By Euclid (related to Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional)
- Proof: By Euclid (related to Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second)
- Proof: By Euclid (related to Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers)
- Proof: By Euclid (related to Proposition: Prop. 9.20: Infinite Number of Primes)
- Proof: By Euclid (related to Proposition: Prop. 9.21: Sum of Even Numbers is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.24: Even Number minus Even Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.25: Even Number minus Odd Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.26: Odd Number minus Odd Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.27: Odd Number minus Even Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even)
- Proof: By Euclid (related to Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half)
- Proof: By Euclid (related to Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double)
- Proof: By Euclid (related to Proposition: Prop. 9.32: Power of Two is Even-Times Even Only)
- Proof: By Euclid (related to Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd)
- Proof: By Euclid (related to Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd)
- Proof: By Euclid (related to Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic)
- Proof: By Euclid (related to Corollary: Cor. 10.111: Thirteen Irrational Straight Lines of Different Order)
- Proof: By Euclid (related to Lemma: Lem. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
- Proof: By Euclid (related to Lemma: Lem. 10.021: Medial is Irrational)
- Proof: By Euclid (related to Lemma: Lem. 10.028.1: Finding Two Squares With Sum Also Square)
- Proof: By Euclid (related to Lemma: Lem. 10.028.2: Finding Two Squares With Sum Not Square)
- Proof: By Euclid (related to Lemma: Lem. 10.032: Constructing Medial Commensurable in Square II)
- Proof: By Euclid (related to Lemma: Lem. 10.041: Side of Sum of Medial Areas is Irrational)
- Proof: By Euclid (related to Lemma: Lem. 10.053: Construction of Rectangle with Area in Mean Proportion to two Square Areas)
- Proof: By Euclid (related to Lemma: Lem. 10.059: Sum of Squares on Unequal Pieces of Segment Is Greater than Twice the Rectangle Contained by Them)
- Proof: By Euclid (related to Lemma: Lem. 10.13: Finding Pythagorean Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number)
- Proof: By Euclid (related to Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm)
- Proof: By Euclid (related to Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio)
- Proof: By Euclid (related to Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.009: Commensurability of Squares)
- Proof: By Euclid (related to Proposition: Prop. 10.010: Construction of Incommensurable Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.012: Commensurability is Transitive Relation)
- Proof: By Euclid (related to Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude)
- Proof: By Euclid (related to Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes)
- Proof: By Euclid (related to Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation)
- Proof: By Euclid (related to Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation)
- Proof: By Euclid (related to Proposition: Prop. 10.019: Product of Rational Numbers is Rational)
- Proof: By Euclid (related to Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational)
- Proof: By Euclid (related to Proposition: Prop. 10.021: Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.022: Square on Medial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial)
- Proof: By Euclid (related to Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial)
- Proof: By Euclid (related to Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square)
- Proof: By Euclid (related to Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area)
- Proof: By Euclid (related to Proposition: Prop. 10.027: Construction of Components of First Bimedial)
- Proof: By Euclid (related to Proposition: Prop. 10.028: Construction of Components of Second Bimedial)
- Proof: By Euclid (related to Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable)
- Proof: By Euclid (related to Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I)
- Proof: By Euclid (related to Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II)
- Proof: By Euclid (related to Proposition: Prop. 10.033: Construction of Components of Major)
- Proof: By Euclid (related to Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area)
- Proof: By Euclid (related to Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas)
- Proof: By Euclid (related to Proposition: Prop. 10.036: Binomial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.037: First Bimedial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.038: Second Bimedial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.039: Major is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely)
- Proof: By Euclid (related to Proposition: Prop. 10.048: Construction of First Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.049: Construction of Second Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.050: Construction of Third Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial)
- Proof: By Euclid (related to Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order)
- Proof: By Euclid (related to Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order)
- Proof: By Euclid (related to Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major)
- Proof: By Euclid (related to Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area)
- Proof: By Euclid (related to Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas)
- Proof: By Euclid (related to Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines)
- Proof: By Euclid (related to Proposition: Prop. 10.073: Apotome is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.074: First Apotome of Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.075: Second Apotome of Medial is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.076: Minor is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational)
- Proof: By Euclid (related to Proposition: Prop. 10.079: Construction of Apotome is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.082: Construction of Minor is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique)
- Proof: By Euclid (related to Proposition: Prop. 10.085: Construction of First Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.086: Construction of Second Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.087: Construction of Third Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.088: Construction of Fourth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.089: Construction of Fifth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.090: Construction of Sixth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.103: Straight Line Commensurable with Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area)
- Proof: By Euclid (related to Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area)
- Proof: By Euclid (related to Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted)
- Proof: By Euclid (related to Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted)
- Proof: By Euclid (related to Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted)
- Proof: By Euclid (related to Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome)
- Proof: By Euclid (related to Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio)
- Proof: By Euclid (related to Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines)
- Proof: By Euclid (related to Lemma: Lem. 11.23: Making a Square Area Equal to the Difference Of Areas of Two Other Incongruent Squares)
- Proof: By Euclid (related to Proposition: 11.02: Two Intersecting Straight Lines are in One Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.01: Straight Line cannot be in Two Planes)
- Proof: By Euclid (related to Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line)
- Proof: By Euclid (related to Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other)
- Proof: By Euclid (related to Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique)
- Proof: By Euclid (related to Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel)
- Proof: By Euclid (related to Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes)
- Proof: By Euclid (related to Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane)
- Proof: By Euclid (related to Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle)
- Proof: By Euclid (related to Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle)
- Proof: By Euclid (related to Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms)
- Proof: By Euclid (related to Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes)
- Proof: By Euclid (related to Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle)
- Proof: By Euclid (related to Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped)
- Proof: By Euclid (related to Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected)
- Proof: By Euclid (related to Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume)
- Proof: By Euclid (related to Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume)
- Proof: By Euclid (related to Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume)
- Proof: By Euclid (related to Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases)
- Proof: By Euclid (related to Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides)
- Proof: By Euclid (related to Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights)
- Proof: By Euclid (related to Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles)
- Proof: By Euclid (related to Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme)
- Proof: By Euclid (related to Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional)
- Proof: By Euclid (related to Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube)
- Proof: By Euclid (related to Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base)
- Proof: By Euclid (related to Corollary: Cor. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Proof: By Euclid (related to Corollary: Cor. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
- Proof: By Euclid (related to Lemma: Lem. 12.02: Areas of Circles are as Squares on Diameters)
- Proof: By Euclid (related to Lemma: Lem. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
- Proof: By Euclid (related to Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters)
- Proof: By Euclid (related to Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters)
- Proof: By Euclid (related to Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms)
- Proof: By Euclid (related to Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms)
- Proof: By Euclid (related to Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra)
- Proof: By Euclid (related to Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides)
- Proof: By Euclid (related to Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights)
- Proof: By Euclid (related to Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height)
- Proof: By Euclid (related to Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases)
- Proof: By Euclid (related to Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis)
- Proof: By Euclid (related to Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights)
- Proof: By Euclid (related to Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights)
- Proof: By Euclid (related to Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles)
- Proof: By Euclid (related to Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres)
- Proof: By Euclid (related to Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters)
- Proof: By Euclid (related to Lemma: Lem. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Lemma: Lem. 13.13: Construction of Regular Tetrahedron within Given Sphere)
- Proof: By Euclid (related to Lemma: Lem. 13.18: Angle of the Pentagon)
- Proof: By Euclid (related to Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment)
- Proof: By Euclid (related to Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome)
- Proof: By Euclid (related to Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal)
- Proof: By Euclid (related to Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio)
- Proof: By Euclid (related to Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa)
- Proof: By Euclid (related to Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor)
- Proof: By Euclid (related to Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle)
- Proof: By Euclid (related to Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.15: Construction of Cube within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere)
- Proof: By Euclid (related to Proposition: Prop. 13.18: There are only Five Platonic Solids)
- Proposition: 1.01: Constructing an Equilateral Triangle
- Proposition: 1.02: Constructing a Segment Equal to an Arbitrary Segment
- Proposition: 1.03: Cutting a Segment at a Given Size
- Proposition: 1.04: "Side-Angle-Side" Theorem for the Congruence of Triangle
- Proposition: 1.05: Isosceles Triangles I
- Proposition: 1.06: Isosceles Triagles II
- Proposition: 1.07: Uniqueness of Triangles
- Proposition: 1.08: "Side-Side-Side" Theorem for the Congruence of Triangles
- Proposition: 1.09: Bisecting an Angle
- Proposition: 1.10: Bisecting a Segment
- Proposition: 1.11: Constructing a Perpendicular Segment to a Straight Line From a Given Point On the Straight Line
- Proposition: 1.12: Constructing a Perpendicular Segment to a Straight Line From a Given Point Not On the Straight Line
- Proposition: 1.13: Angles at Intersections of Straight Lines
- Proposition: 1.14: Combining Rays to Straight Lines
- Proposition: 1.15: Opposite Angles on Intersecting Straight Lines
- Proposition: 1.16: The Exterior Angle Is Greater Than Either of the Non-Adjacent Interior Angles
- Proposition: 1.17: The Sum of Two Angles of a Triangle
- Proposition: 1.18: Angles and Sides in a Triangle I
- Proposition: 1.19: Angles and Sides in a Triangle II
- Proposition: 1.20: The Sum of the Lengths of Any Pair of Sides of a Triangle (Triangle Inequality)
- Proposition: 1.21: Triangles within Triangles
- Proposition: 1.22: Construction of Triangles From Arbitrary Segments
- Proposition: 1.23: Constructing an Angle Equal to an Arbitrary Rectilinear Angle
- Proposition: 1.24: Angles and Sides in a Triangle III
- Proposition: 1.25: Angles and Sides in a Triangle IV
- Proposition: 1.26: "Angle-Side-Angle" and "Angle-Angle-Side" Theorems for the Congruence of Triangles
- Proposition: 1.27: Parallel Lines I
- Proposition: 1.28: Parallel Lines II
- Proposition: 1.29: Parallel Lines III
- Proposition: 1.30: Transitivity of Parallel Lines
- Proposition: 1.31: Constructing a Parallel Line from a Line and a Point
- Proposition: 1.32: Sum Of Angles in a Triangle and Exterior Angle
- Proposition: 1.33: Parallel Equal Segments Determine a Parallelogram
- Proposition: 1.34: Opposite Sides and Opposite Angles of Parallelograms
- Proposition: 1.35: Parallelograms On the Same Base and On the Same Parallels
- Proposition: 1.36: Parallelograms on Equal Bases and on the Same Parallels
- Proposition: 1.37: Triangles of Equal Area I
- Proposition: 1.38: Triangles of Equal Area II
- Proposition: 1.39: Triangles of Equal Area III
- Proposition: 1.40: Triangles of Equal Area IV
- Proposition: 1.41: Parallelograms and Triagles
- Proposition: 1.42: Construction of Parallelograms I
- Proposition: 1.43: Complementary Segments of Parallelograms
- Proposition: 1.44: Construction of Parallelograms II
- Proposition: 1.45: Construction of Parallelograms III
- Proposition: 1.46: Construction of a Square on a Given Segment
- Proposition: 1.47: Pythagorean Theorem
- Proposition: 1.48: The Converse of the Pythagorean Theorem
- Proposition: 11.02: Two Intersecting Straight Lines are in One Plane
- Proposition: 2.01: Summing Areas or Rectangles
- Proposition: 2.02: Square is Sum of Two Rectangles
- Proposition: 2.03: Rectangle is Sum of Square and Rectangle
- Proposition: 2.04: Square of Sum
- Proposition: 2.05: Rectangle is Difference of Two Squares
- Proposition: 2.06: Square of Sum with One Halved Summand
- Proposition: 2.07: Sum of Squares
- Proposition: 2.08: Square of Sum with One Doubled Summand
- Proposition: 2.09: Sum of Squares of Sum and Difference
- Proposition: 2.10: Sum of Squares (Half)
- Proposition: 2.12: Law of Cosines (for Obtuse Angles)
- Proposition: 2.13: Law of Cosines (for Acute Angles)
- Proposition: 2.14: Constructing a Square from a Rectilinear Figure
- Proposition: 3.01: Finding the Center of a given Circle
- Proposition: 3.02: Chord Lies Inside its Circle
- Proposition: 3.03: Conditions for Diameter to be a Perpendicular Bisector
- Proposition: 3.04: Chords do not Bisect Each Other
- Proposition: 3.05: Intersecting Circles have Different Centers
- Proposition: 3.06: Touching Circles have Different Centers
- Proposition: 3.07: Relative Lengths of Lines Inside Circle
- Proposition: 3.08: Relative Lengths of Lines Outside Circle
- Proposition: 3.09: Condition for Point to be Center of Circle
- Proposition: 3.10: Two Circles have at most Two Points of Intersection
- Proposition: 3.11: Line Joining Centers of Two Circles Touching Internally
- Proposition: 3.12: Line Joining Centers of Two Circles Touching Externally
- Proposition: 3.13: Circles Touch at One Point at Most
- Proposition: 3.14: Equal Chords in Circle
- Proposition: 3.15: Relative Lengths of Chords of Circles
- Proposition: 3.16: Line at Right Angles to Diameter of Circle At its Ends Touches the Circle
- Proposition: 3.17: Construction of Tangent from Point to Circle
- Proposition: 3.18: Radius at Right Angle to Tangent
- Proposition: 3.19: Right Angle to Tangent of Circle Goes Through Center
- Proposition: 3.20: Inscribed Angle Theorem
- Proposition: 3.21: Angles in Same Segment of Circle are Equal
- Proposition: 3.22: Opposite Angles of Cyclic Quadrilateral
- Proposition: 3.23: Segment on Given Base Unique
- Proposition: 3.24: Similar Segments on Equal Bases are Equal
- Proposition: 3.25: Construction of Circle from Segment
- Proposition: 3.26: Equal Angles and Arcs in Equal Circles
- Proposition: 3.27: Angles on Equal Arcs are Equal
- Proposition: 3.28: Straight Lines Cut Off Equal Arcs in Equal Circles
- Proposition: 3.29: Equal Arcs of Circles Subtended by Equal Straight Lines
- Proposition: 3.30: Bisection of Arc
- Proposition: 3.31: Relative Sizes of Angles in Segments
- Proposition: 3.32: Angles made by Chord with Tangent
- Proposition: 3.33: Construction of Segment on Given Line Admitting Given Angle
- Proposition: 3.34: Construction of Segment on Given Circle Admitting Given Angle
- Proposition: 3.35: Intersecting Chord Theorem
- Proposition: 3.36: Tangent Secant Theorem
- Proposition: 3.37: Converse of Tangent Secant Theorem
- Proposition: 4.01: Fitting Chord Into Circle
- Proposition: 4.02: Inscribing in Circle Triangle Equiangular with Given Angles
- Proposition: 4.03: Circumscribing about Circle Triangle Equiangular with Given Angles
- Proposition: 4.04: Inscribing Circle in Triangle
- Proposition: 4.05: Circumscribing Circle about Triangle
- Proposition: 4.06: Inscribing Square in Circle
- Proposition: 4.07: Circumscribing Square about Circle
- Proposition: 4.08: Inscribing Circle in Square
- Proposition: 4.09: Circumscribing Circle about Square
- Proposition: 4.10: Construction of Isosceles Triangle whose Base Angle is Twice Apex
- Proposition: 4.11: Inscribing Regular Pentagon in Circle
- Proposition: 4.12: Circumscribing Regular Pentagon about Circle
- Proposition: 4.13: Inscribing Circle in Regular Pentagon
- Proposition: 4.14: Circumscribing Circle about Regular Pentagon
- Proposition: 4.15: Side of Hexagon Inscribed in a Circle Equals the Radius of that Circle
- Proposition: 4.16: Inscribing Regular Pentakaidecagon in Circle
- Proposition: 5.02: Multiplication of Numbers is Right Distributive over Addition
- Proposition: 5.03: Multiplication of Numbers is Associative
- Proposition: 5.05: Multiplication of Real Numbers is Left Distributive over Subtraction
- Proposition: 5.06: Multiplication of Real Numbers is Right Distributive over Subtraction
- Proposition: 5.07: Ratios of Equal Magnitudes
- Proposition: 5.08: Relative Sizes of Ratios on Unequal Magnitudes
- Proposition: 5.09: Magnitudes with Same Ratios are Equal
- Proposition: 5.10: Relative Sizes of Magnitudes on Unequal Ratios
- Proposition: 5.11: Equality of Ratios is Transitive
- Proposition: 5.12: Sum of Components of Equal Ratios
- Proposition: 5.13: Relative Sizes of Proportional Magnitudes
- Proposition: 5.14: Relative Sizes of Components of Ratios
- Proposition: 5.15: Ratio Equals its Multiples
- Proposition: 5.16: Proportional Magnitudes are Proportional Alternately
- Proposition: 5.17: Magnitudes Proportional Compounded are Proportional Separated
- Proposition: 5.18: Magnitudes Proportional Separated are Proportional Compounded
- Proposition: 5.19: Proportional Magnitudes have Proportional Remainders
- Proposition: 5.20: Relative Sizes of Successive Ratios
- Proposition: 5.21: Relative Sizes of Elements in Perturbed Proportion
- Proposition: 5.22: Equality of Ratios Ex Aequali
- Proposition: 5.23: Equality of Ratios in Perturbed Proportion
- Proposition: 5.24: Sum of Antecedents of Proportion
- Proposition: 5.25: Sum of Antecedent and Consequent of Proportion
- Proposition: 6.01: Areas of Triangles and Parallelograms Proportional to Base
- Proposition: 6.02: Parallel Line in Triangle Cuts Sides Proportionally
- Proposition: 6.03: Angle Bisector Theorem
- Proposition: 6.04: Equiangular Triangles are Similar
- Proposition: 6.05: Triangles with Proportional Sides are Similar
- Proposition: 6.06: Triangles with One Equal Angle and Two Sides Proportional are Similar
- Proposition: 6.07: Triangles with One Equal Angle and Two Other Sides Proportional are Similar
- Proposition: 6.08: Perpendicular in Right-Angled Triangle makes two Similar Triangles
- Proposition: 6.09: Construction of Part of Line
- Proposition: 6.10: Construction of Similarly Cut Straight Line
- Proposition: 6.11: Construction of Segment in Squared Ratio
- Proposition: 6.12: Construction of Fourth Proportional Straight Line
- Proposition: 6.13: Construction of Mean Proportional
- Proposition: 6.14: Characterization of Congruent Parallelograms
- Proposition: 6.15: Characterization of Congruent Triangles
- Proposition: 6.16: Rectangles Contained by Proportional Straight Lines
- Proposition: 6.17: Rectangles Contained by Three Proportional Straight Lines
- Proposition: 6.18: Construction of Similar Polygon
- Proposition: 6.20: Similar Polygons are Composed of Similar Triangles
- Proposition: 6.21: Similarity of Polygons is Transitive
- Proposition: 6.22: Similar Figures on Proportional Straight Lines
- Proposition: 6.23: Ratio of Areas of Equiangular Parallelograms
- Proposition: 6.24: Parallelograms About Diameter are Similar
- Proposition: 6.25: Construction of Figure Similar to One and Equal to Another
- Proposition: 6.26: Parallelogram Similar and in Same Angle has Same Diameter
- Proposition: 6.27: Similar Parallelogram on Half a Straight Line
- Proposition: 6.28: Construction of Parallelogram Equal to Given Figure Less a Parallelogram
- Proposition: 6.29: Construction of Parallelogram Equal to Given Figure Exceeding a Parallelogram
- Proposition: 6.30: Construction of the Inverse Golden Section
- Proposition: 6.31: Similar Figures on Sides of Right-Angled Triangle
- Proposition: 6.32: Triangles with Two Sides Parallel and Equal
- Proposition: 6.33: Angles in Circles have Same Ratio as Arcs
- Proposition: 7.01: Sufficient Condition for Coprimality
- Proposition: 7.02: Greatest Common Divisor of Two Numbers - Euclidean Algorithm
- Proposition: 7.04: Smaller Numbers are Dividing or not Dividing Larger Numbers
- Proposition: 7.05: Divisors Obey Distributive Law (Sum)
- Proposition: 7.06: Division with Quotient and Remainder Obeys Distributive Law (Sum)
- Proposition: 7.07: Divisors Obey Distributive Law (Difference)
- Proposition: 7.08: Division with Quotient and Remainder Obeys Distributivity Law (Difference)
- Proposition: 7.09: Alternate Ratios of Equal Fractions
- Proposition: 7.10: Multiples of Alternate Ratios of Equal Fractions
- Proposition: 7.11: Proportional Numbers have Proportional Differences
- Proposition: 7.12: Ratios of Numbers is Distributive over Addition
- Proposition: 7.13: Proportional Numbers are Proportional Alternately
- Proposition: 7.14: Proportion of Numbers is Transitive
- Proposition: 7.15: Alternate Ratios of Multiples
- Proposition: 7.16: Natural Number Multiplication is Commutative
- Proposition: 7.17: Multiples of Ratios of Numbers
- Proposition: 7.18: Ratios of Multiples of Numbers
- Proposition: 7.19: Relation of Ratios to Products
- Proposition: 7.20: Ratios of Fractions in Lowest Terms
- Proposition: 7.21: Co-prime Numbers form Fraction in Lowest Terms
- Proposition: 7.22: Numbers forming Fraction in Lowest Terms are Co-prime
- Proposition: 7.23: Divisor of One of Co-prime Numbers is Co-prime to Other
- Proposition: 7.24: Integer Co-prime to all Factors is Co-prime to Whole
- Proposition: 7.25: Square of Co-prime Number is Co-prime
- Proposition: 7.26: Product of Co-prime Pairs is Co-prime
- Proposition: 7.27: Powers of Co-prime Numbers are Co-prime
- Proposition: 7.28: Numbers are Co-prime iff Sum is Co-prime to Both
- Proposition: 7.29: Prime not Divisor implies Co-prime
- Proposition: 7.30: Euclidean Lemma
- Proposition: 7.31: Existence of Prime Divisors
- Proposition: 7.32: Natural Number is Prime or has Prime Factor
- Proposition: 7.33: Least Ratio of Numbers
- Proposition: 7.34: Existence of Least Common Multiple
- Proposition: 7.35: Least Common Multiple Divides Common Multiple
- Proposition: 7.36: Least Common Multiple of Three Numbers
- Proposition: 7.37: Integer Divided by Divisor is Integer
- Proposition: 7.38: Divisor is Reciprocal of Divisor of Integer
- Proposition: 7.39: Least Number with Three Given Fractions
- Proposition: 8.01: Geometric Progression with Co-prime Extremes is in Lowest Terms
- Proposition: 8.02: Construction of Geometric Progression in Lowest Terms
- Proposition: 8.03: Geometric Progression in Lowest Terms has Co-prime Extremes
- Proposition: 8.04: Construction of Sequence of Numbers with Given Ratios
- Proposition: 8.05: Ratio of Products of Sides of Plane Numbers
- Proposition: 8.06: First Element of Geometric Progression not dividing Second
- Proposition: 8.07: First Element of Geometric Progression that divides Last also divides Second
- Proposition: 8.08: Geometric Progressions in Proportion have Same Number of Elements
- Proposition: 9.36: Theorem of Even Perfect Numbers (First Part)
- Proposition: Prop. 10.001: Existence of Fraction of Number Smaller than Given Number
- Proposition: Prop. 10.002: Incommensurable Magnitudes do not Terminate in Euclidean Algorithm
- Proposition: Prop. 10.003: Greatest Common Measure of Commensurable Magnitudes
- Proposition: Prop. 10.004: Greatest Common Measure of Three Commensurable Magnitudes
- Proposition: Prop. 10.005: Ratio of Commensurable Magnitudes
- Proposition: Prop. 10.006: Magnitudes with Rational Ratio are Commensurable
- Proposition: Prop. 10.007: Incommensurable Magnitudes Have Irrational Ratio
- Proposition: Prop. 10.008: Magnitudes with Irrational Ratio are Incommensurable
- Proposition: Prop. 10.009: Commensurability of Squares
- Proposition: Prop. 10.010: Construction of Incommensurable Lines
- Proposition: Prop. 10.011: Commensurability of Elements of Proportional Magnitudes
- Proposition: Prop. 10.012: Commensurability is Transitive Relation
- Proposition: Prop. 10.013: Commensurable Magnitudes are Incommensurable with Same Magnitude
- Proposition: Prop. 10.014: Commensurability of Squares on Proportional Straight Lines
- Proposition: Prop. 10.015: Commensurability of Sum of Commensurable Magnitudes
- Proposition: Prop. 10.016: Incommensurability of Sum of Incommensurable Magnitudes
- Proposition: Prop. 10.017: Condition for Commensurability of Roots of Quadratic Equation
- Proposition: Prop. 10.018: Condition for Incommensurability of Roots of Quadratic Equation
- Proposition: Prop. 10.019: Product of Rational Numbers is Rational
- Proposition: Prop. 10.020: Quotient of Rational Numbers is Rational
- Proposition: Prop. 10.021: Medial is Irrational
- Proposition: Prop. 10.022: Square on Medial Straight Line
- Proposition: Prop. 10.023: Segment Commensurable with Medial Segment is Medial
- Proposition: Prop. 10.024: Rectangle Contained by Medial Straight Lines Commensurable in Length is Medial
- Proposition: Prop. 10.025: Rationality of Rectangle Contained by Medial Straight Lines Commensurable in Square
- Proposition: Prop. 10.026: Medial Area not greater than Medial Area by Rational Area
- Proposition: Prop. 10.027: Construction of Components of First Bimedial
- Proposition: Prop. 10.028: Construction of Components of Second Bimedial
- Proposition: Prop. 10.029: Construction of Rational Straight Lines Commensurable in Square When Square Differences Commensurable
- Proposition: Prop. 10.030: Construction of Rational Straight Lines Commensurable in Square Only When Square Differences Incommensurable
- Proposition: Prop. 10.031: Constructing Medial Commensurable in Square I
- Proposition: Prop. 10.032: Constructing Medial Commensurable in Square II
- Proposition: Prop. 10.033: Construction of Components of Major
- Proposition: Prop. 10.034: Construction of Components of Side of Rational plus Medial Area
- Proposition: Prop. 10.035: Construction of Components of Side of Sum of Medial Areas
- Proposition: Prop. 10.036: Binomial is Irrational
- Proposition: Prop. 10.038: Second Bimedial is Irrational
- Proposition: Prop. 10.039: Major is Irrational
- Proposition: Prop. 10.040: Side of Rational plus Medial Area is Irrational
- Proposition: Prop. 10.041: Side of Sum of Medial Areas is Irrational
- Proposition: Prop. 10.042: Binomial Straight Line is Divisible into Terms Uniquely
- Proposition: Prop. 10.043: First Bimedial Straight Line is Divisible Uniquely
- Proposition: Prop. 10.044: Second Bimedial Straight Line is Divisible Uniquely
- Proposition: Prop. 10.045: Major Straight Line is Divisible Uniquely
- Proposition: Prop. 10.046: Side of Rational Plus Medial Area is Divisible Uniquely
- Proposition: Prop. 10.047: Side of Sum of Two Medial Areas is Divisible Uniquely
- Proposition: Prop. 10.048: Construction of First Binomial Straight Line
- Proposition: Prop. 10.049: Construction of Second Binomial Straight Line
- Proposition: Prop. 10.050: Construction of Third Binomial Straight Line
- Proposition: Prop. 10.051: Construction of Fourth Binomial Straight Line
- Proposition: Prop. 10.052: Construction of Fifth Binomial Straight Line
- Proposition: Prop. 10.053: Construction of Sixth Binomial Straight Line
- Proposition: Prop. 10.054: Root of Area contained by Rational Straight Line and First Binomial
- Proposition: Prop. 10.055: Root of Area contained by Rational Straight Line and Second Binomial
- Proposition: Prop. 10.056: Root of Area contained by Rational Straight Line and Third Binomial
- Proposition: Prop. 10.057: Root of Area contained by Rational Straight Line and Fourth Binomial
- Proposition: Prop. 10.058: Root of Area contained by Rational Straight Line and Fifth Binomial
- Proposition: Prop. 10.059: Root of Area contained by Rational Straight Line and Sixth Binomial
- Proposition: Prop. 10.060: Square on Binomial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.061: Square on First Bimedial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.062: Square on Second Bimedial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.063: Square on Major Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.064: Square on Side of Rational plus Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.065: Square on Side of Sum of two Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.066: Straight Line Commensurable with Binomial Straight Line is Binomial and of Same Order
- Proposition: Prop. 10.067: Straight Line Commensurable with Bimedial Straight Line is Bimedial and of Same Order
- Proposition: Prop. 10.068: Straight Line Commensurable with Major Straight Line is Major
- Proposition: Prop. 10.069: Straight Line Commensurable with Side of Rational plus Medial Area
- Proposition: Prop. 10.070: Straight Line Commensurable with Side of Sum of two Medial Areas
- Proposition: Prop. 10.071: Sum of Rational Area and Medial Area gives rise to four Irrational Straight Lines
- Proposition: Prop. 10.072: Sum of two Incommensurable Medial Areas give rise to two Irrational Straight Lines
- Proposition: Prop. 10.073: Apotome is Irrational
- Proposition: Prop. 10.074: First Apotome of Medial is Irrational
- Proposition: Prop. 10.075: Second Apotome of Medial is Irrational
- Proposition: Prop. 10.076: Minor is Irrational
- Proposition: Prop. 10.077: That which produces Medial Whole with Rational Area is Irrational
- Proposition: Prop. 10.078: That which produces Medial Whole with Medial Area is Irrational
- Proposition: Prop. 10.079: Construction of Apotome is Unique
- Proposition: Prop. 10.080: Construction of First Apotome of Medial is Unique
- Proposition: Prop. 10.081: Construction of Second Apotome of Medial is Unique
- Proposition: Prop. 10.082: Construction of Minor is Unique
- Proposition: Prop. 10.083: Construction of that which produces Medial Whole with Rational Area is Unique
- Proposition: Prop. 10.084: Construction of that which produces Medial Whole with Medial Area is Unique
- Proposition: Prop. 10.085: Construction of First Apotome
- Proposition: Prop. 10.086: Construction of Second Apotome
- Proposition: Prop. 10.087: Construction of Third Apotome
- Proposition: Prop. 10.088: Construction of Fourth Apotome
- Proposition: Prop. 10.089: Construction of Fifth Apotome
- Proposition: Prop. 10.090: Construction of Sixth Apotome
- Proposition: Prop. 10.091: Side of Area Contained by Rational Straight Line and First Apotome
- Proposition: Prop. 10.092: Side of Area Contained by Rational Straight Line and Second Apotome
- Proposition: Prop. 10.093: Side of Area Contained by Rational Straight Line and Third Apotome
- Proposition: Prop. 10.094: Side of Area Contained by Rational Straight Line and Fourth Apotome
- Proposition: Prop. 10.095: Side of Area Contained by Rational Straight Line and Fifth Apotome
- Proposition: Prop. 10.096: Side of Area Contained by Rational Straight Line and Sixth Apotome
- Proposition: Prop. 10.097: Square on Apotome applied to Rational Straight Line
- Proposition: Prop. 10.098: Square on First Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.099: Square on Second Apotome of Medial Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.100: Square on Minor Straight Line applied to Rational Straight Line
- Proposition: Prop. 10.101: Square on Straight Line which produces Medial Whole with Rational Area applied to Rational Straight Line
- Proposition: Prop. 10.102: Square on Straight Line which produces Medial Whole with Medial Area applied to Rational Straight Line
- Proposition: Prop. 10.103: Straight Line Commensurable with Apotome
- Proposition: Prop. 10.104: Straight Line Commensurable with Apotome of Medial Straight Line
- Proposition: Prop. 10.105: Straight Line Commensurable with Minor Straight Line
- Proposition: Prop. 10.106: Straight Line Commensurable with that which produces Medial Whole with Rational Area
- Proposition: Prop. 10.107: Straight Line Commensurable With That Which Produces Medial Whole With Medial Area
- Proposition: Prop. 10.108: Side of Remaining Area from Rational Area from which Medial Area Subtracted
- Proposition: Prop. 10.109: Two Irrational Straight Lines arising from Medial Area from which Rational Area Subtracted
- Proposition: Prop. 10.110: Two Irrational Straight Lines arising from Medial Area from which Medial Area Subtracted
- Proposition: Prop. 10.111: Apotome not same with Binomial Straight Line
- Proposition: Prop. 10.112: Square on Rational Straight Line applied to Binomial Straight Line
- Proposition: Prop. 10.113: Square on Rational Straight Line applied to Apotome
- Proposition: Prop. 10.114: Area contained by Apotome and Binomial Straight Line Commensurable with Terms of Apotome and in same Ratio
- Proposition: Prop. 10.115: From Medial Straight Line arises Infinite Number of Irrational Straight Lines
- Proposition: Prop. 11.01: Straight Line cannot be in Two Planes
- Proposition: Prop. 11.03: Common Section of Two Planes is Straight Line
- Proposition: Prop. 11.04: Line Perpendicular to Two Intersecting Lines is Perpendicular to their Plane
- Proposition: Prop. 11.05: Three Intersecting Lines Perpendicular to Another Line are in One Plane
- Proposition: Prop. 11.06: Two Lines Perpendicular to Same Plane are Parallel
- Proposition: Prop. 11.07: Line joining Points on Parallel Lines is in Same Plane
- Proposition: Prop. 11.08: Line Parallel to Perpendicular Line to Plane is Perpendicular to Same Plane
- Proposition: Prop. 11.09: Lines Parallel to Same Line not in Same Plane are Parallel to each other
- Proposition: Prop. 11.10: Two Lines Meeting which are Parallel to Two Other Lines Meeting contain Equal Angles
- Proposition: Prop. 11.11: Construction of Straight Line Perpendicular to Plane from point not on Plane
- Proposition: Prop. 11.12: Construction of Straight Line Perpendicular to Plane from point on Plane
- Proposition: Prop. 11.13: Straight Line Perpendicular to Plane from Point is Unique
- Proposition: Prop. 11.14: Planes Perpendicular to same Straight Line are Parallel
- Proposition: Prop. 11.15: Planes through Parallel Pairs of Meeting Lines are Parallel
- Proposition: Prop. 11.16: Common Sections of Parallel Planes with other Plane are Parallel
- Proposition: Prop. 11.17: Straight Lines cut in Same Ratio by Parallel Planes
- Proposition: Prop. 11.18: Plane through Straight Line Perpendicular to other Plane is Perpendicular to that Plane
- Proposition: Prop. 11.19: Common Section of Planes Perpendicular to other Plane is Perpendicular to that Plane
- Proposition: Prop. 11.20: Sum of Two Angles of Three containing Solid Angle is Greater than Other Angle
- Proposition: Prop. 11.21: Solid Angle contained by Plane Angles is Less than Four Right Angles
- Proposition: Prop. 11.22: Extremities of Line Segments containing three Plane Angles any Two of which are Greater than Other form Triangle
- Proposition: Prop. 11.23: Sum of Plane Angles Used to Construct a Solid Angle is Less Than Four Right Angles
- Proposition: Prop. 11.24: Opposite Planes of Solid contained by Parallel Planes are Equal Parallelograms
- Proposition: Prop. 11.25: Parallelepiped cut by Plane Parallel to Opposite Planes
- Proposition: Prop. 11.26: Construction of Solid Angle equal to Given Solid Angle
- Proposition: Prop. 11.27: Construction of Parallelepiped Similar to Given Parallelepiped
- Proposition: Prop. 11.28: Parallelepiped cut by Plane through Diagonals of Opposite Planes is Bisected
- Proposition: Prop. 11.29: Parallelepipeds on Same Base and Same Height whose Extremities are on Same Lines are Equal in Volume
- Proposition: Prop. 11.30: Parallelepipeds on Same Base and Same Height whose Extremities are not on Same Lines are Equal in Volume
- Proposition: Prop. 11.31: Parallelepipeds on Equal Bases and Same Height are Equal in Volume
- Proposition: Prop. 11.32: Parallelepipeds of Same Height have Volume Proportional to Bases
- Proposition: Prop. 11.33: Volumes of Similar Parallelepipeds are in Triplicate Ratio to Length of Corresponding Sides
- Proposition: Prop. 11.34: Parallelepipeds are of Equal Volume iff Bases are in Reciprocal Proportion to Heights
- Proposition: Prop. 11.35: Condition for Equal Angles contained by Elevated Straight Lines from Plane Angles
- Proposition: Prop. 11.36: Parallelepiped formed from Three Proportional Lines equal to Equilateral Parallelepiped with Equal Angles to it forme
- Proposition: Prop. 11.37: Four Straight Lines are Proportional iff Similar Parallelepipeds formed on them are Proportional
- Proposition: Prop. 11.38: Common Section of Bisecting Planes of Cube Bisect and are Bisected by Diagonal of Cube
- Proposition: Prop. 11.39: Prisms of Equal Height with Parallelogram and Triangle as Base
- Proposition: Prop. 12.01: Areas of Similar Polygons Inscribed in Circles are as Squares on Diameters
- Proposition: Prop. 12.02: Areas of Circles are as Squares on Diameters
- Proposition: Prop. 12.03: Tetrahedron divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition: Prop. 12.04: Proportion of Sizes of Tetrahedra divided into Two Similar Tetrahedra and Two Equal Prisms
- Proposition: Prop. 12.05: Sizes of Tetrahedra of Same Height are as Bases
- Proposition: Prop. 12.06: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases
- Proposition: Prop. 12.07: Prism on Triangular Base divided into Three Equal Tetrahedra
- Proposition: Prop. 12.08: Volumes of Similar Tetrahedra are in Cubed Ratio of Corresponding Sides
- Proposition: Prop. 12.09: Tetrahedra are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition: Prop. 12.10: Volume of Cone is Third of Cylinder on Same Base and of Same Height
- Proposition: Prop. 12.11: Volume of Cones or Cylinders of Same Height are in Same Ratio as Bases
- Proposition: Prop. 12.12: Volumes of Similar Cones and Cylinders are in Triplicate Ratio of Diameters of Bases
- Proposition: Prop. 12.13: Volumes of Parts of Cylinder cut by Plane Parallel to Opposite Planes are as Parts of Axis
- Proposition: Prop. 12.14: Volumes of Cones or Cylinders on Equal Bases are in Same Ratio as Heights
- Proposition: Prop. 12.15: Cones or Cylinders are Equal iff Bases are Reciprocally Proportional to Heights
- Proposition: Prop. 12.16: Construction of Equilateral Polygon with Even Number of Sides in Outer of Concentric Circles
- Proposition: Prop. 12.17: Construction of Polyhedron in Outer of Concentric Spheres
- Proposition: Prop. 12.18: Volumes of Spheres are in Triplicate Ratio of Diameters
- Proposition: Prop. 13.01: Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.02: Converse of Area of Square on Greater Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.03: Area of Square on Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.04: Area of Squares on Whole and Lesser Segment of Straight Line cut in Extreme and Mean Ratio
- Proposition: Prop. 13.05: Straight Line cut in Extreme and Mean Ratio plus its Greater Segment
- Proposition: Prop. 13.06: Segments of Rational Straight Line cut in Extreme and Mean Ratio are Apotome
- Proposition: Prop. 13.07: Equilateral Pentagon is Equiangular if Three Angles are Equal
- Proposition: Prop. 13.08: Straight Lines Subtending Two Consecutive Angles in Regular Pentagon cut in Extreme and Mean Ratio
- Proposition: Prop. 13.09: Sides Appended of Hexagon and Decagon inscribed in same Circle are cut in Extreme and Mean Ratio
- Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa
- Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor
- Proposition: Prop. 13.12: Square on Side of Equilateral Triangle inscribed in Circle is Triple Square on Radius of Circle
- Proposition: Prop. 13.13: Construction of Regular Tetrahedron within Given Sphere
- Proposition: Prop. 13.14: Construction of Regular Octahedron within Given Sphere
- Proposition: Prop. 13.15: Construction of Cube within Given Sphere
- Proposition: Prop. 13.16: Construction of Regular Icosahedron within Given Sphere
- Proposition: Prop. 13.17: Construction of Regular Dodecahedron within Given Sphere
- Proposition: Prop. 13.18: There are only Five Platonic Solids
- Proposition: Prop. 8.09: Elements of Geometric Progression between Co-prime Numbers
- Proposition: Prop. 8.10: Product of Geometric Progressions from One
- Proposition: Prop. 8.11: Between two Squares exists one Mean Proportional
- Proposition: Prop. 8.12: Between two Cubes exist two Mean Proportionals
- Proposition: Prop. 8.13: Powers of Elements of Geometric Progression are in Geometric Progression
- Proposition: Prop. 8.14: Number divides Number iff Square divides Square
- Proposition: Prop. 8.15: Number divides Number iff Cube divides Cube
- Proposition: Prop. 8.16: Number does not divide Number iff Square does not divide Square
- Proposition: Prop. 8.17: Number does not divide Number iff Cube does not divide Cube
- Proposition: Prop. 8.18: Between two Similar Plane Numbers exists one Mean Proportional
- Proposition: Prop. 8.19: Between two Similar Solid Numbers exist two Mean Proportionals
- Proposition: Prop. 8.20: Numbers between which exists one Mean Proportional are Similar Plane
- Proposition: Prop. 8.21: Numbers between which exist two Mean Proportionals are Similar Solid
- Proposition: Prop. 8.22: If First of Three Numbers in Geometric Progression is Square then Third is Square
- Proposition: Prop. 8.23: If First of Four Numbers in Geometric Progression is Cube then Fourth is Cube
- Proposition: Prop. 8.24: If Ratio of Square to Number is as between Two Squares then Number is Square
- Proposition: Prop. 8.25: If Ratio of Cube to Number is as between Two Cubes then Number is Cube
- Proposition: Prop. 8.26: Similar Plane Numbers have Same Ratio as between Two Squares
- Proposition: Prop. 8.27: Similar Solid Numbers have Same Ratio as between Two Cubes
- Proposition: Prop. 9.01: Product of Similar Plane Numbers is Square
- Proposition: Prop. 9.02: Numbers whose Product is Square are Similar Plane Numbers
- Proposition: Prop. 9.03: Square of Cube Number is Cube
- Proposition: Prop. 9.04: Cube Number multiplied by Cube Number is Cube
- Proposition: Prop. 9.05: Number multiplied by Cube Number making Cube is itself Cube
- Proposition: Prop. 9.06: Number Squared making Cube is itself Cube
- Proposition: Prop. 9.07: Product of Composite Number with Number is Solid Number
- Proposition: Prop. 9.08: Elements of Geometric Progression from One which are Powers of Number
- Proposition: Prop. 9.09: Elements of Geometric Progression from One where First Element is Power of Number
- Proposition: Prop. 9.10: Elements of Geometric Progression from One where First Element is not Power of Number
- Proposition: Prop. 9.11: Elements of Geometric Progression from One which Divide Later Elements
- Proposition: Prop. 9.12: Elements of Geometric Progression from One Divisible by Prime
- Proposition: Prop. 9.13: Divisibility of Elements of Geometric Progression from One where First Element is Prime
- Proposition: Prop. 9.15: Sum of Pair of Elements of Geometric Progression with Three Elements in Lowest Terms is Co-prime to other Element
- Proposition: Prop. 9.16: Two Co-prime Integers have no Third Integer Proportional
- Proposition: Prop. 9.17: Last Element of Geometric Progression with Co-prime Extremes has no Integer Proportional as First to Second
- Proposition: Prop. 9.18: Condition for Existence of Third Number Proportional to Two Numbers
- Proposition: Prop. 9.19: Condition for Existence of Fourth Number Proportional to Three Numbers
- Proposition: Prop. 9.20: Infinite Number of Primes
- Proposition: Prop. 9.21: Sum of Even Numbers is Even
- Proposition: Prop. 9.22: Sum of Even Number of Odd Numbers is Even
- Proposition: Prop. 9.23: Sum of Odd Number of Odd Numbers is Odd
- Proposition: Prop. 9.24: Even Number minus Even Number is Even
- Proposition: Prop. 9.25: Even Number minus Odd Number is Odd
- Proposition: Prop. 9.26: Odd Number minus Odd Number is Even
- Proposition: Prop. 9.27: Odd Number minus Even Number is Odd
- Proposition: Prop. 9.28: Odd Number multiplied by Even Number is Even
- Proposition: Prop. 9.29: Odd Number multiplied by Odd Number is Odd
- Proposition: Prop. 9.30: Odd Divisor of Even Number Also Divides Its Half
- Proposition: Prop. 9.31: Odd Number Co-prime to Number is also Co-prime to its Double
- Proposition: Prop. 9.32: Power of Two is Even-Times Even Only
- Proposition: Prop. 9.33: Number whose Half is Odd is Even-Times Odd
- Proposition: Prop. 9.34: Number neither whose Half is Odd nor Power of Two is both Even-Times Even and Even-Times Odd
- Section: Book 01: Fundamentals of Plane Geometry Involving Straight Lines
- Section: Book 05: Proportion
- Section: Book 06: Similar Figures
- Section: Book 07: Elementary Number Theory
- Section: Book 08: Continued Proportion
- Section: Book 09: Applications of Number Theory
- Section: Book 10: Incommensurable Magnitudes
- Section: Book 12: Proportional Stereometry
- Section: Book 13: Platonic Solids
- Theorem: Prop. 9.14: Fundamental Theorem of Arithmetic
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